alpha_beta_pruning/alpha_beta.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Alpha-beta pruning \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Alpha-beta pruning* serves as an enhancement to the *minimax* algorithm, which is commonly employed in the realm of two-person games like tic-tac-toe, checkers, chess, and Go, where game trees are a central component of strategy evaluation.\n",
    "\n",
    "A game tree comprehensively encompasses all potential moves and outcomes within a game, and it can often be exceptionally vast or even theoretically infinite in scope. Each distinct game state corresponds to a node within this tree structure, with the root node representing the initial state. As players make their moves, each possible play traces a path or branch within the game tree, ultimately culminating in a terminal node, signifying the conclusion of the game.\n",
    "\n",
    "To determine the optimal move in this complex web of possibilities, it is essential to employ an evaluation function capable of ascribing *values* to these nodes. The player who takes the first turn aspires to reach a terminal node with the highest value, while their opponent seeks to achieve the opposite objective. This back-and-forth dynamic continues as the players alternate their moves in the pursuit of strategic victory.\n",
    "\n",
    "Moving forward, we'll designate the player taking the first move as the *maximizer*, while the other player will be referred to as the *minimizer*.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "try:\n",
    "  from alpha_beta import * # type: ignore\n",
    "except ImportError:\n",
    "  url = 'https://raw.githubusercontent.com/jhjeong314/CS_Math/main/alpha_beta_pruning'\n",
    "  import httpimport\n",
    "  with httpimport.remote_repo(url):\n",
    "    from alpha_beta import * # type: ignore"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's consider a very simple game for illustrative purposes, which is depicted in the diagram below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"166pt\" height=\"122pt\"\n",
       " viewBox=\"0.00 0.00 165.80 122.40\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 118.4)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-118.4 161.8,-118.4 161.8,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"78.4\" cy=\"-100\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"78.4\" y=\"-95.72\" font-family=\"Times,serif\" font-size=\"12.00\">0</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"57.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"57.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">3</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M72.21,-86.98C69.55,-81.81 66.44,-75.77 63.76,-70.56\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"100.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"100.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">2</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M84.88,-86.98C87.72,-81.71 91.05,-75.55 93.89,-70.28\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"14.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"14.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">1</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M47.43,-46.74C40.58,-40.24 31.52,-31.64 24.62,-25.09\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"57.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"57.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">4</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M57.4,-42.39C57.4,-38.13 57.4,-33.45 57.4,-29.2\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"100.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"100.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">2</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M100.4,-42.39C100.4,-38.13 100.4,-33.45 100.4,-29.2\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"143.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"143.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">3</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M110.37,-46.74C117.22,-40.24 126.28,-31.64 133.18,-25.09\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107dd3f90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dot_data = '''graph {\n",
    "  node [width=.4 height=.4 fixedsize=true fontsize=12];\n",
    "  nodesep=0.2;\n",
    "  ranksep=0.2;\n",
    "\n",
    "  root [label=\"0\"];\n",
    "  0 [label=\"3\"];\n",
    "  00 [label=\"1\"];\n",
    "  01 [label=\"4\"];\n",
    "  1 [label=\"2\"];\n",
    "  10 [label=\"2\"];\n",
    "  11 [label=\"3\"];\n",
    "\n",
    "  root -- {0 1};\n",
    "  0 -- {00 01};\n",
    "  1 -- {10 11};\n",
    "}\n",
    "'''\n",
    "graph = graphviz.Source(dot_data)\n",
    "display(graph)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this graphical representation, the numbers associated with each node represent their respective values. To facilitate our discussion, we can assign a unique tree-index to each node according to the following scheme:\n",
    "\n",
    "- The root node: identified as an empty string.\n",
    "- The left child of the root: identified as 0.\n",
    "- The right child of the root: identified as 1.\n",
    "- The left child of the node with tree-index 0: identified as 00.\n",
    "- The right child of the node with tree-index 0: identified as 01.\n",
    "- The left child of the node with tree-index 1: identified as 10.\n",
    "- The right child of the node with tree-index 1: identified as 11.\n",
    "\n",
    "It is clear that this systematic method allows us to assign tree-indices to all nodes within an arbitrarily extensive tree structure."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. Minimax algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, for the first move, the maximizer faces a choice between node 0 (with a value of 3) and node 1 (with a value of 2).\n",
    "\n",
    "If the maximizer opts for the larger value of 3, the minimizer will undoubtedly select node 00, which has a value of 1. However, should the maximizer go for the initially smaller value of 2, the minimizer will seize node 10, possessing a value of 2, which is bigger than the other terminal value 1.\n",
    "\n",
    "Therefore, the optimal decision for the maximizer is to select node 1 (with a value of 2). This choice aligns with the maximizer's overarching strategy, which involves maximizing the minimum value among its potential moves—hence the term *minimax*.\n",
    "\n",
    "This principle holds true not only for game trees with a height of 2 but, encouragingly, extends to game trees of arbitrary height. We can substantiate this fact rigorously by employing an inductive approach based on the tree's height."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's essential to emphasize that, within this strategy, the values assigned to internal nodes are inconsequential. Only the values at terminal nodes hold significance. To facilitate our comprehension of the search algorithm, we will operate under the assumption that we can readily evaluate the value at any node, even those located at a considerable distance from the root."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In our second example, the game tree has a depth of 3. Consequently, the maximizer has the opportunity to make two moves, whereas the minimizer makes only one move."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"338pt\" height=\"165pt\"\n",
       " viewBox=\"0.00 0.00 337.80 165.20\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 161.2)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-161.2 333.8,-161.2 333.8,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"164.4\" cy=\"-142.8\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"164.4\" y=\"-138.53\" font-family=\"Times,serif\" font-size=\"12.00\">Max</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121.4\" cy=\"-100\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"121.4\" y=\"-95.72\" font-family=\"Times,serif\" font-size=\"12.00\">Min</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M154.43,-132.34C147.58,-125.84 138.52,-117.24 131.62,-110.69\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"207.4\" cy=\"-100\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"207.4\" y=\"-95.72\" font-family=\"Times,serif\" font-size=\"12.00\">Min</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M174.37,-132.34C181.22,-125.84 190.28,-117.24 197.18,-110.69\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"57.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"57.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">Max</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M109.36,-91.33C97.92,-84.03 80.81,-73.12 69.38,-65.84\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"121.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">Max</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M121.4,-85.19C121.4,-80.93 121.4,-76.25 121.4,-72\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"14.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"14.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">4</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M47.43,-46.74C40.58,-40.24 31.52,-31.64 24.62,-25.09\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"57.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"57.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">8</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M57.4,-42.39C57.4,-38.13 57.4,-33.45 57.4,-29.2\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"100.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"100.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">9</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M115.21,-44.18C112.55,-39.01 109.44,-32.97 106.76,-27.76\"/>\n",
       "</g>\n",
       "<!-- 011 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"143.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"143.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">3</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;011 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>01&#45;&#45;011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M127.88,-44.18C130.72,-38.91 134.05,-32.75 136.89,-27.48\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"207.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"207.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">Max</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M207.4,-85.19C207.4,-80.93 207.4,-76.25 207.4,-72\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"272.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"272.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">Max</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M219.32,-91.52C231.05,-84.16 248.85,-72.98 260.55,-65.64\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"186.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"186.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">2</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201.21,-44.18C198.55,-39.01 195.44,-32.97 192.76,-27.76\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"229.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"229.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;2</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M213.88,-44.18C216.72,-38.91 220.05,-32.75 222.89,-27.48\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"272.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"272.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">9</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M272.4,-42.39C272.4,-38.13 272.4,-33.45 272.4,-29.2\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"315.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"315.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;1</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M282.37,-46.74C289.22,-40.24 298.28,-31.64 305.18,-25.09\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107dd7a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "list1 = [4,8,9,3,2,-2,9,-1]\n",
    "root = build_bin_tree(list1)\n",
    "root.draw_tree() # same as root.draw_tree(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As illustrated in the diagram above, the values are initially assigned to the terminal nodes. Meanwhile, for the internal nodes, those attributed to the maximizer are labeled as `Max`, while those assigned to the minimizer are denoted as `Min`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(8, [0, 0, 1])\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"338pt\" height=\"165pt\"\n",
       " viewBox=\"0.00 0.00 337.80 165.20\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 161.2)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-161.2 333.8,-161.2 333.8,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"164.4\" cy=\"-142.8\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"164.4\" y=\"-138.53\" font-family=\"Times,serif\" font-size=\"12.00\">8</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121.4\" cy=\"-100\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"121.4\" y=\"-95.72\" font-family=\"Times,serif\" font-size=\"12.00\">8</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M154.43,-132.34C147.58,-125.84 138.52,-117.24 131.62,-110.69\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"207.4\" cy=\"-100\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"207.4\" y=\"-95.72\" font-family=\"Times,serif\" font-size=\"12.00\">2</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M174.37,-132.34C181.22,-125.84 190.28,-117.24 197.18,-110.69\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"57.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"57.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">8</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M109.36,-91.33C97.92,-84.03 80.81,-73.12 69.38,-65.84\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"121.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">9</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M121.4,-85.19C121.4,-80.93 121.4,-76.25 121.4,-72\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"14.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"14.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">4</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M47.43,-46.74C40.58,-40.24 31.52,-31.64 24.62,-25.09\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"57.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"57.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">8</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M57.4,-42.39C57.4,-38.13 57.4,-33.45 57.4,-29.2\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"100.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"100.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">9</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M115.21,-44.18C112.55,-39.01 109.44,-32.97 106.76,-27.76\"/>\n",
       "</g>\n",
       "<!-- 011 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"143.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"143.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">3</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;011 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>01&#45;&#45;011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M127.88,-44.18C130.72,-38.91 134.05,-32.75 136.89,-27.48\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"207.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"207.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">2</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M207.4,-85.19C207.4,-80.93 207.4,-76.25 207.4,-72\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"272.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"272.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">9</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M219.32,-91.52C231.05,-84.16 248.85,-72.98 260.55,-65.64\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"186.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"186.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">2</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201.21,-44.18C198.55,-39.01 195.44,-32.97 192.76,-27.76\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"229.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"229.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;2</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M213.88,-44.18C216.72,-38.91 220.05,-32.75 222.89,-27.48\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"272.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"272.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">9</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M272.4,-42.39C272.4,-38.13 272.4,-33.45 272.4,-29.2\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"315.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"315.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;1</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M282.37,-46.74C289.22,-40.24 298.28,-31.64 305.18,-25.09\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107de4f50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# When we apply the minimax algorithm to above, we get the following result:\n",
    "\n",
    "print(minimax(root, True))\n",
    "root.draw_tree()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The maximizer achieves a result of 8 through the minimax algorithm. The *path*, which represents the sequence of moves leading to this result, can be expressed as [0, 0, 1]. To succinctly convey this information, the text output is presented as (8, [0, 0, 1]).\n",
    "\n",
    "In our implementation, we traverse the tree nodes in postorder because we can evaluate a node only after all of its children have been evaluated.\n",
    "\n",
    "Node 00 is evaluated as 8 because max(4,8) = 8.  Node 01 is evaluated as 9 because max(9,3) = 9.  Node 0 is evaluated as 8 because Min(8,9) = 8. We can continue this way to evaluate all nodes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"338pt\" height=\"165pt\"\n",
       " viewBox=\"0.00 0.00 337.80 165.20\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 161.2)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-161.2 333.8,-161.2 333.8,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"164.4\" cy=\"-142.8\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"164.4\" y=\"-138.53\" font-family=\"Times,serif\" font-size=\"12.00\">Min</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121.4\" cy=\"-100\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"121.4\" y=\"-95.72\" font-family=\"Times,serif\" font-size=\"12.00\">Max</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M154.43,-132.34C147.58,-125.84 138.52,-117.24 131.62,-110.69\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"207.4\" cy=\"-100\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"207.4\" y=\"-95.72\" font-family=\"Times,serif\" font-size=\"12.00\">Max</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M174.37,-132.34C181.22,-125.84 190.28,-117.24 197.18,-110.69\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"57.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"57.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">Min</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M109.36,-91.33C97.92,-84.03 80.81,-73.12 69.38,-65.84\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"121.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">Min</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M121.4,-85.19C121.4,-80.93 121.4,-76.25 121.4,-72\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"14.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"14.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">4</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M47.43,-46.74C40.58,-40.24 31.52,-31.64 24.62,-25.09\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"57.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"57.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">8</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M57.4,-42.39C57.4,-38.13 57.4,-33.45 57.4,-29.2\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"100.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"100.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">9</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M115.21,-44.18C112.55,-39.01 109.44,-32.97 106.76,-27.76\"/>\n",
       "</g>\n",
       "<!-- 011 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"143.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"143.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">3</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;011 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>01&#45;&#45;011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M127.88,-44.18C130.72,-38.91 134.05,-32.75 136.89,-27.48\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"207.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"207.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">Min</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M207.4,-85.19C207.4,-80.93 207.4,-76.25 207.4,-72\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"272.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"272.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">Min</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M219.32,-91.52C231.05,-84.16 248.85,-72.98 260.55,-65.64\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"186.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"186.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">2</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201.21,-44.18C198.55,-39.01 195.44,-32.97 192.76,-27.76\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"229.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"229.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;2</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M213.88,-44.18C216.72,-38.91 220.05,-32.75 222.89,-27.48\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"272.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"272.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">9</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M272.4,-42.39C272.4,-38.13 272.4,-33.45 272.4,-29.2\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"315.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"315.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;1</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M282.37,-46.74C289.22,-40.24 298.28,-31.64 305.18,-25.09\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107de6490>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Let's assume that the minimizer moves first.\n",
    "# Well, this is against my previous assumption that the maximizer \n",
    "# moves first. But it may helps us understand the algorithm better.\n",
    "\n",
    "list1 = [4,8,9,3,2,-2,9,-1]\n",
    "root = build_bin_tree(list1)\n",
    "root.draw_tree(False) # False means the minimizer moves first"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(-1, [1, 1, 1])\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"338pt\" height=\"165pt\"\n",
       " viewBox=\"0.00 0.00 337.80 165.20\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 161.2)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-161.2 333.8,-161.2 333.8,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"164.4\" cy=\"-142.8\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"164.4\" y=\"-138.53\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;1</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121.4\" cy=\"-100\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"121.4\" y=\"-95.72\" font-family=\"Times,serif\" font-size=\"12.00\">4</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M154.43,-132.34C147.58,-125.84 138.52,-117.24 131.62,-110.69\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"207.4\" cy=\"-100\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"207.4\" y=\"-95.72\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;1</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M174.37,-132.34C181.22,-125.84 190.28,-117.24 197.18,-110.69\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"57.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"57.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">4</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M109.36,-91.33C97.92,-84.03 80.81,-73.12 69.38,-65.84\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"121.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">3</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M121.4,-85.19C121.4,-80.93 121.4,-76.25 121.4,-72\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"14.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"14.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">4</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M47.43,-46.74C40.58,-40.24 31.52,-31.64 24.62,-25.09\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"57.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"57.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">8</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M57.4,-42.39C57.4,-38.13 57.4,-33.45 57.4,-29.2\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"100.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"100.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">9</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M115.21,-44.18C112.55,-39.01 109.44,-32.97 106.76,-27.76\"/>\n",
       "</g>\n",
       "<!-- 011 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"143.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"143.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">3</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;011 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>01&#45;&#45;011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M127.88,-44.18C130.72,-38.91 134.05,-32.75 136.89,-27.48\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"207.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"207.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;2</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M207.4,-85.19C207.4,-80.93 207.4,-76.25 207.4,-72\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"272.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"272.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;1</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M219.32,-91.52C231.05,-84.16 248.85,-72.98 260.55,-65.64\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"186.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"186.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">2</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201.21,-44.18C198.55,-39.01 195.44,-32.97 192.76,-27.76\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"229.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"229.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;2</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M213.88,-44.18C216.72,-38.91 220.05,-32.75 222.89,-27.48\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"272.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"272.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">9</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M272.4,-42.39C272.4,-38.13 272.4,-33.45 272.4,-29.2\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"315.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"315.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;1</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M282.37,-46.74C289.22,-40.24 298.28,-31.64 305.18,-25.09\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107de6c90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# In this case, the minimax algorithm will return the following result:\n",
    "\n",
    "print(minimax(root, False))\n",
    "root.draw_tree(False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "While the minimax algorithm is a powerful strategy for game analysis, it may face limitations when applied to real-world problems with a high branching factor, such as 10, and a game tree height of 16. \n",
    "\n",
    "To put this into perspective, evaluating situations in such a scenario would require processing a staggering 10^16 nodes. Even with a high-performance computer capable of evaluating one million nodes per second, the computational task would extend over a daunting 10^10 seconds, equivalent to several hundred years.\n",
    "\n",
    "This immense computational demand highlights the need for more efficient algorithms and heuristics to tackle complex problems with substantial branching factors and deep game trees.\n",
    "\n",
    "FYI, in the game of chess, the typical branching factor, representing the number of possible moves in a given position, is approximately 14. Skilled chess players, including world chess masters, often possess the ability to think ahead by as many as 15 moves, allowing them to plan and strategize several steps into the future."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. Alpha-beta pruning algorithm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(8, [0, 0, 1])\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"338pt\" height=\"165pt\"\n",
       " viewBox=\"0.00 0.00 337.80 165.20\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 161.2)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-161.2 333.8,-161.2 333.8,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"164.4\" cy=\"-142.8\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"164.4\" y=\"-138.53\" font-family=\"Times,serif\" font-size=\"12.00\">8</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121.4\" cy=\"-100\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"121.4\" y=\"-95.72\" font-family=\"Times,serif\" font-size=\"12.00\">8</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M154.43,-132.34C147.58,-125.84 138.52,-117.24 131.62,-110.69\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"207.4\" cy=\"-100\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"207.4\" y=\"-95.72\" font-family=\"Times,serif\" font-size=\"12.00\">2</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M174.37,-132.34C181.22,-125.84 190.28,-117.24 197.18,-110.69\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"57.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"57.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">8</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M109.36,-91.33C97.92,-84.03 80.81,-73.12 69.38,-65.84\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"121.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">9</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M121.4,-85.19C121.4,-80.93 121.4,-76.25 121.4,-72\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"14.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"14.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">4</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M47.43,-46.74C40.58,-40.24 31.52,-31.64 24.62,-25.09\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"57.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"57.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">8</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M57.4,-42.39C57.4,-38.13 57.4,-33.45 57.4,-29.2\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"100.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"100.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">9</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M115.21,-44.18C112.55,-39.01 109.44,-32.97 106.76,-27.76\"/>\n",
       "</g>\n",
       "<!-- 011 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"143.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"143.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\" fill=\"red\">Min</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;011 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>01&#45;&#45;011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M127.88,-44.18C130.72,-38.91 134.05,-32.75 136.89,-27.48\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"207.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"207.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">2</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M207.4,-85.19C207.4,-80.93 207.4,-76.25 207.4,-72\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"272.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"272.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\" fill=\"red\">Max</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M219.32,-91.52C231.05,-84.16 248.85,-72.98 260.55,-65.64\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"186.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"186.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">2</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201.21,-44.18C198.55,-39.01 195.44,-32.97 192.76,-27.76\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"229.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"229.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;2</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M213.88,-44.18C216.72,-38.91 220.05,-32.75 222.89,-27.48\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"272.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"272.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\" fill=\"red\">Min</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M272.4,-42.39C272.4,-38.13 272.4,-33.45 272.4,-29.2\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"315.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"315.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\" fill=\"red\">Min</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M282.37,-46.74C289.22,-40.24 298.28,-31.64 305.18,-25.09\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107de79d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "root = build_bin_tree(list1)\n",
    "print(alpha_beta_pruning(root, True))\n",
    "root.draw_tree()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nodes within pruned branches do not require evaluation, conserving computational resources. These nodes are labeled as 'Max' for maximizing nodes and 'Min' for minimizing nodes. Additionally, they are visually represented in red.\n",
    "\n",
    "Allow me to clarify why there's no need to evaluate node 011. Node 0 functions as a minimizing node and will be assigned the minimum of 8 and the value of node 01, which is yet to be determined. Once we ascertain that the value of node 010 is 9, it becomes evident that node 01, a maximizing node, will be assigned a value greater than or equal to 9. Consequently, we can conclusively determine that node 0 will be assigned the value 8 without requiring knowledge of the value at node 011. In other words, the evaluation of node 011 does not impact the outcome for node 0. This elucidates why there is no need to evaluate node 011.\n",
    "\n",
    "At this point, I'm confident that you can explain why there's no need to evaluate node 11 without much difficulty."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This concept can be incorporated to develop an algorithm for systematically identifying the nodes that do not require evaluation.\n",
    "\n",
    "Throughout the computation, two values, `alpha` and `beta`, are employed to represent the guaranteed minimum score for the maximizing player and the guaranteed maximum score for the minimizing player, respectively. Initially, `alpha` is set to negative infinity, and `beta` is initialized to positive infinity. One could say that both players start with their worst possible scores.\n",
    "\n",
    "Maximizing node updates `alpha` (from elder children) while minimizing node updates `beta` (from elder children too). These values are passed to the younger children to determine whether further search is necessary.\n",
    "\n",
    "As an maximizer node evaluates its child nodes, if it discovers a node whose value exceeds or equals the `beta` threshold, there is no necessity to explore its younger siblings. This is because the maximizer's parent node is a minimizer and thus it will not use the current node's value. Further search can only make the value of this maximizer node larger and won't be used the minimizer parent anyway. \n",
    "\n",
    "Likewise, when the minimizer assesses its child nodes and finds a node whose value is less than or equal to the `alpha` threshold, there is no need to delve deeper into the descendants of that node, as they too will not affect the course of the game."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the above example, we observed that we saved 4 out of a total of 15 nodes. Well this doesn't look like an impressive result. What's more, in the next example where the minimizer makes the first move, there are no pruned branches at all! Such a poor result was obtained because our game tree was tiny."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(-1, [1, 1, 1])\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"338pt\" height=\"165pt\"\n",
       " viewBox=\"0.00 0.00 337.80 165.20\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 161.2)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-161.2 333.8,-161.2 333.8,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"164.4\" cy=\"-142.8\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"164.4\" y=\"-138.53\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;1</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121.4\" cy=\"-100\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"121.4\" y=\"-95.72\" font-family=\"Times,serif\" font-size=\"12.00\">4</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M154.43,-132.34C147.58,-125.84 138.52,-117.24 131.62,-110.69\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"207.4\" cy=\"-100\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"207.4\" y=\"-95.72\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;1</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M174.37,-132.34C181.22,-125.84 190.28,-117.24 197.18,-110.69\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"57.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"57.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">4</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M109.36,-91.33C97.92,-84.03 80.81,-73.12 69.38,-65.84\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"121.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">3</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M121.4,-85.19C121.4,-80.93 121.4,-76.25 121.4,-72\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"14.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"14.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">4</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M47.43,-46.74C40.58,-40.24 31.52,-31.64 24.62,-25.09\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"57.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"57.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">8</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M57.4,-42.39C57.4,-38.13 57.4,-33.45 57.4,-29.2\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"100.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"100.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">9</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M115.21,-44.18C112.55,-39.01 109.44,-32.97 106.76,-27.76\"/>\n",
       "</g>\n",
       "<!-- 011 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"143.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"143.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">3</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;011 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>01&#45;&#45;011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M127.88,-44.18C130.72,-38.91 134.05,-32.75 136.89,-27.48\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"207.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"207.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;2</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M207.4,-85.19C207.4,-80.93 207.4,-76.25 207.4,-72\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"272.4\" cy=\"-57.2\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"272.4\" y=\"-52.92\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;1</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M219.32,-91.52C231.05,-84.16 248.85,-72.98 260.55,-65.64\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"186.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"186.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">2</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201.21,-44.18C198.55,-39.01 195.44,-32.97 192.76,-27.76\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"229.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"229.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;2</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M213.88,-44.18C216.72,-38.91 220.05,-32.75 222.89,-27.48\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"272.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"272.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">9</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M272.4,-42.39C272.4,-38.13 272.4,-33.45 272.4,-29.2\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"315.4\" cy=\"-14.4\" rx=\"14.4\" ry=\"14.4\"/>\n",
       "<text text-anchor=\"middle\" x=\"315.4\" y=\"-10.12\" font-family=\"Times,serif\" font-size=\"12.00\">&#45;1</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M282.37,-46.74C289.22,-40.24 298.28,-31.64 305.18,-25.09\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107dd7690>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "root = build_bin_tree(list1)\n",
    "print(alpha_beta_pruning(root, False))\n",
    "root.draw_tree(False) # no pruning occurs in this case"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The computation time for the minimax algorithm can be readily understood as being in $O(b^d)$, where $b$ represents the branching factor and $d$ denotes the search depth.\n",
    "\n",
    "It is known that the computation time for the alpha-beta pruning algorithm is typically in $O(b^{d/2})$ on average. As an example, when $b = 10$ and $d = 20$, the computation time decreases by a factor of 10 billion!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. More examples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"506pt\" height=\"172pt\"\n",
       " viewBox=\"0.00 0.00 506.00 172.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 168)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-168 502,-168 502,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-153.2\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-149.32\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"185\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"185\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M256.44,-148.6C240.96,-142.1 208.52,-128.48 193.26,-122.07\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"313\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"313\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M272.5,-146.95C281.46,-140.67 296.38,-130.23 305.39,-123.92\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"89\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"89\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M176.48,-113.62C158.31,-107.26 115.92,-92.42 97.64,-86.02\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"185\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"185\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M185,-106.34C185,-102.17 185,-97.36 185,-93.2\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M81.5,-75.75C72.54,-69.47 57.62,-59.03 48.61,-52.72\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"89\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"89\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M89,-70.74C89,-66.57 89,-61.76 89,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0000 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>0000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"9\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"9\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0000 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>000&#45;&#45;0000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.52,-38.6C28.97,-32.77 21.02,-24.42 15.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0001 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>0001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0001 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>000&#45;&#45;0001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M41,-35.14C41,-30.97 41,-26.16 41,-22\"/>\n",
       "</g>\n",
       "<!-- 0010 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>0010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"73\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"73\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0010 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>001&#45;&#45;0010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M84.8,-36.57C82.43,-31.59 79.48,-25.4 77.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0011 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>0011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"105\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"105\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0011 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>001&#45;&#45;0011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M93.2,-36.57C95.57,-31.59 98.52,-25.4 100.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M180.8,-72.17C178.43,-67.19 175.48,-61 173.12,-56.05\"/>\n",
       "</g>\n",
       "<!-- 011 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;011 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>01&#45;&#45;011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M189.2,-72.17C191.57,-67.19 194.52,-61 196.88,-56.05\"/>\n",
       "</g>\n",
       "<!-- 0100 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>0100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"137\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"137\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0100 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>010&#45;&#45;0100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M162.52,-38.6C156.97,-32.77 149.02,-24.42 143.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0101 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>0101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0101 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>010&#45;&#45;0101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M169,-35.14C169,-30.97 169,-26.16 169,-22\"/>\n",
       "</g>\n",
       "<!-- 0110 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>0110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 011&#45;&#45;0110 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>011&#45;&#45;0110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-35.14C201,-30.97 201,-26.16 201,-22\"/>\n",
       "</g>\n",
       "<!-- 0111 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>0111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"233\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"233\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 011&#45;&#45;0111 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>011&#45;&#45;0111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M207.48,-38.6C213.03,-32.77 220.98,-24.42 226.53,-18.59\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"313\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"313\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M313,-106.34C313,-102.17 313,-97.36 313,-93.2\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"409\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"409\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M321.52,-113.62C339.69,-107.26 382.08,-92.42 400.36,-86.02\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M308.8,-72.17C306.43,-67.19 303.48,-61 301.12,-56.05\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M317.2,-72.17C319.57,-67.19 322.52,-61 324.88,-56.05\"/>\n",
       "</g>\n",
       "<!-- 1000 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>1000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1000 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>100&#45;&#45;1000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M290.52,-38.6C284.97,-32.77 277.02,-24.42 271.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 1001 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>1001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1001 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>100&#45;&#45;1001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M297,-35.14C297,-30.97 297,-26.16 297,-22\"/>\n",
       "</g>\n",
       "<!-- 1010 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>1010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1010 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>101&#45;&#45;1010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M329,-35.14C329,-30.97 329,-26.16 329,-22\"/>\n",
       "</g>\n",
       "<!-- 1011 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>1011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"361\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"361\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1011 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>101&#45;&#45;1011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M335.48,-38.6C341.03,-32.77 348.98,-24.42 354.53,-18.59\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"409\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"409\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M409,-70.74C409,-66.57 409,-61.76 409,-57.6\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M416.5,-75.75C425.46,-69.47 440.38,-59.03 449.39,-52.72\"/>\n",
       "</g>\n",
       "<!-- 1100 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>1100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"393\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"393\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1100 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>110&#45;&#45;1100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M404.8,-36.57C402.43,-31.59 399.48,-25.4 397.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 1101 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>1101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1101 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>110&#45;&#45;1101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M413.2,-36.57C415.57,-31.59 418.52,-25.4 420.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 1110 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>1110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1110 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>111&#45;&#45;1110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-35.14C457,-30.97 457,-26.16 457,-22\"/>\n",
       "</g>\n",
       "<!-- 1111 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>1111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"489\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"489\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1111 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>111&#45;&#45;1111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M463.48,-38.6C469.03,-32.77 476.98,-24.42 482.53,-18.59\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107de6450>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "list2 = [8, 7, 3, 9, 9, 8, 2, 4, 1, 5, 8, 9, 9, 9, 3, 4]\n",
    "root_2 = build_bin_tree(list2)\n",
    "root_2.draw_tree(True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"506pt\" height=\"172pt\"\n",
       " viewBox=\"0.00 0.00 506.00 172.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 168)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-168 502,-168 502,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-153.2\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-149.32\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"185\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"185\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M256.44,-148.6C240.96,-142.1 208.52,-128.48 193.26,-122.07\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"313\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"313\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M272.5,-146.95C281.46,-140.67 296.38,-130.23 305.39,-123.92\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"89\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"89\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M176.48,-113.62C158.31,-107.26 115.92,-92.42 97.64,-86.02\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"185\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"185\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M185,-106.34C185,-102.17 185,-97.36 185,-93.2\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M81.5,-75.75C72.54,-69.47 57.62,-59.03 48.61,-52.72\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"89\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"89\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M89,-70.74C89,-66.57 89,-61.76 89,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0000 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>0000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"9\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"9\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0000 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>000&#45;&#45;0000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.52,-38.6C28.97,-32.77 21.02,-24.42 15.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0001 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>0001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0001 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>000&#45;&#45;0001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M41,-35.14C41,-30.97 41,-26.16 41,-22\"/>\n",
       "</g>\n",
       "<!-- 0010 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>0010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"73\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"73\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0010 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>001&#45;&#45;0010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M84.8,-36.57C82.43,-31.59 79.48,-25.4 77.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0011 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>0011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"105\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"105\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0011 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>001&#45;&#45;0011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M93.2,-36.57C95.57,-31.59 98.52,-25.4 100.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M180.8,-72.17C178.43,-67.19 175.48,-61 173.12,-56.05\"/>\n",
       "</g>\n",
       "<!-- 011 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;011 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>01&#45;&#45;011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M189.2,-72.17C191.57,-67.19 194.52,-61 196.88,-56.05\"/>\n",
       "</g>\n",
       "<!-- 0100 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>0100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"137\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"137\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0100 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>010&#45;&#45;0100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M162.52,-38.6C156.97,-32.77 149.02,-24.42 143.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0101 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>0101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0101 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>010&#45;&#45;0101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M169,-35.14C169,-30.97 169,-26.16 169,-22\"/>\n",
       "</g>\n",
       "<!-- 0110 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>0110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 011&#45;&#45;0110 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>011&#45;&#45;0110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-35.14C201,-30.97 201,-26.16 201,-22\"/>\n",
       "</g>\n",
       "<!-- 0111 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>0111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"233\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"233\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 011&#45;&#45;0111 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>011&#45;&#45;0111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M207.48,-38.6C213.03,-32.77 220.98,-24.42 226.53,-18.59\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"313\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"313\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M313,-106.34C313,-102.17 313,-97.36 313,-93.2\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"409\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"409\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M321.52,-113.62C339.69,-107.26 382.08,-92.42 400.36,-86.02\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M308.8,-72.17C306.43,-67.19 303.48,-61 301.12,-56.05\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M317.2,-72.17C319.57,-67.19 322.52,-61 324.88,-56.05\"/>\n",
       "</g>\n",
       "<!-- 1000 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>1000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1000 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>100&#45;&#45;1000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M290.52,-38.6C284.97,-32.77 277.02,-24.42 271.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 1001 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>1001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1001 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>100&#45;&#45;1001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M297,-35.14C297,-30.97 297,-26.16 297,-22\"/>\n",
       "</g>\n",
       "<!-- 1010 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>1010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1010 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>101&#45;&#45;1010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M329,-35.14C329,-30.97 329,-26.16 329,-22\"/>\n",
       "</g>\n",
       "<!-- 1011 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>1011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"361\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"361\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1011 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>101&#45;&#45;1011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M335.48,-38.6C341.03,-32.77 348.98,-24.42 354.53,-18.59\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"409\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"409\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M409,-70.74C409,-66.57 409,-61.76 409,-57.6\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M416.5,-75.75C425.46,-69.47 440.38,-59.03 449.39,-52.72\"/>\n",
       "</g>\n",
       "<!-- 1100 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>1100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"393\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"393\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1100 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>110&#45;&#45;1100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M404.8,-36.57C402.43,-31.59 399.48,-25.4 397.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 1101 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>1101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1101 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>110&#45;&#45;1101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M413.2,-36.57C415.57,-31.59 418.52,-25.4 420.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 1110 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>1110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1110 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>111&#45;&#45;1110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-35.14C457,-30.97 457,-26.16 457,-22\"/>\n",
       "</g>\n",
       "<!-- 1111 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>1111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"489\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"489\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1111 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>111&#45;&#45;1111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M463.48,-38.6C469.03,-32.77 476.98,-24.42 482.53,-18.59\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107df4f90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "root_2.draw_tree(False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(8, [1, 0, 1, 0])\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"506pt\" height=\"172pt\"\n",
       " viewBox=\"0.00 0.00 506.00 172.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 168)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-168 502,-168 502,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-153.2\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-149.32\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"185\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"185\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M256.44,-148.6C240.96,-142.1 208.52,-128.48 193.26,-122.07\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"313\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"313\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M272.5,-146.95C281.46,-140.67 296.38,-130.23 305.39,-123.92\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"89\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"89\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M176.48,-113.62C158.31,-107.26 115.92,-92.42 97.64,-86.02\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"185\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"185\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M185,-106.34C185,-102.17 185,-97.36 185,-93.2\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M81.5,-75.75C72.54,-69.47 57.62,-59.03 48.61,-52.72\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"89\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"89\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M89,-70.74C89,-66.57 89,-61.76 89,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0000 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>0000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"9\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"9\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0000 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>000&#45;&#45;0000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.52,-38.6C28.97,-32.77 21.02,-24.42 15.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0001 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>0001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0001 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>000&#45;&#45;0001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M41,-35.14C41,-30.97 41,-26.16 41,-22\"/>\n",
       "</g>\n",
       "<!-- 0010 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>0010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"73\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"73\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0010 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>001&#45;&#45;0010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M84.8,-36.57C82.43,-31.59 79.48,-25.4 77.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0011 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>0011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"105\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"105\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0011 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>001&#45;&#45;0011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M93.2,-36.57C95.57,-31.59 98.52,-25.4 100.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M180.8,-72.17C178.43,-67.19 175.48,-61 173.12,-56.05\"/>\n",
       "</g>\n",
       "<!-- 011 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;011 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>01&#45;&#45;011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M189.2,-72.17C191.57,-67.19 194.52,-61 196.88,-56.05\"/>\n",
       "</g>\n",
       "<!-- 0100 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>0100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"137\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"137\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0100 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>010&#45;&#45;0100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M162.52,-38.6C156.97,-32.77 149.02,-24.42 143.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0101 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>0101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0101 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>010&#45;&#45;0101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M169,-35.14C169,-30.97 169,-26.16 169,-22\"/>\n",
       "</g>\n",
       "<!-- 0110 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>0110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 011&#45;&#45;0110 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>011&#45;&#45;0110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-35.14C201,-30.97 201,-26.16 201,-22\"/>\n",
       "</g>\n",
       "<!-- 0111 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>0111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"233\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"233\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 011&#45;&#45;0111 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>011&#45;&#45;0111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M207.48,-38.6C213.03,-32.77 220.98,-24.42 226.53,-18.59\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"313\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"313\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M313,-106.34C313,-102.17 313,-97.36 313,-93.2\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"409\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"409\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M321.52,-113.62C339.69,-107.26 382.08,-92.42 400.36,-86.02\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M308.8,-72.17C306.43,-67.19 303.48,-61 301.12,-56.05\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M317.2,-72.17C319.57,-67.19 322.52,-61 324.88,-56.05\"/>\n",
       "</g>\n",
       "<!-- 1000 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>1000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1000 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>100&#45;&#45;1000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M290.52,-38.6C284.97,-32.77 277.02,-24.42 271.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 1001 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>1001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1001 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>100&#45;&#45;1001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M297,-35.14C297,-30.97 297,-26.16 297,-22\"/>\n",
       "</g>\n",
       "<!-- 1010 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>1010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1010 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>101&#45;&#45;1010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M329,-35.14C329,-30.97 329,-26.16 329,-22\"/>\n",
       "</g>\n",
       "<!-- 1011 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>1011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"361\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"361\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1011 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>101&#45;&#45;1011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M335.48,-38.6C341.03,-32.77 348.98,-24.42 354.53,-18.59\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"409\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"409\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M409,-70.74C409,-66.57 409,-61.76 409,-57.6\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M416.5,-75.75C425.46,-69.47 440.38,-59.03 449.39,-52.72\"/>\n",
       "</g>\n",
       "<!-- 1100 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>1100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"393\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"393\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1100 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>110&#45;&#45;1100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M404.8,-36.57C402.43,-31.59 399.48,-25.4 397.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 1101 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>1101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1101 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>110&#45;&#45;1101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M413.2,-36.57C415.57,-31.59 418.52,-25.4 420.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 1110 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>1110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1110 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>111&#45;&#45;1110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-35.14C457,-30.97 457,-26.16 457,-22\"/>\n",
       "</g>\n",
       "<!-- 1111 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>1111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"489\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"489\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1111 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>111&#45;&#45;1111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M463.48,-38.6C469.03,-32.77 476.98,-24.42 482.53,-18.59\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107df5090>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(minimax(root_2, True))\n",
    "root_2.draw_tree()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(5, [1, 0, 0, 1])\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"506pt\" height=\"172pt\"\n",
       " viewBox=\"0.00 0.00 506.00 172.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 168)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-168 502,-168 502,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-153.2\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-149.32\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"185\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"185\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M256.44,-148.6C240.96,-142.1 208.52,-128.48 193.26,-122.07\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"313\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"313\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M272.5,-146.95C281.46,-140.67 296.38,-130.23 305.39,-123.92\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"89\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"89\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M176.48,-113.62C158.31,-107.26 115.92,-92.42 97.64,-86.02\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"185\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"185\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M185,-106.34C185,-102.17 185,-97.36 185,-93.2\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M81.5,-75.75C72.54,-69.47 57.62,-59.03 48.61,-52.72\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"89\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"89\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M89,-70.74C89,-66.57 89,-61.76 89,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0000 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>0000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"9\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"9\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0000 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>000&#45;&#45;0000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.52,-38.6C28.97,-32.77 21.02,-24.42 15.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0001 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>0001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0001 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>000&#45;&#45;0001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M41,-35.14C41,-30.97 41,-26.16 41,-22\"/>\n",
       "</g>\n",
       "<!-- 0010 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>0010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"73\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"73\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0010 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>001&#45;&#45;0010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M84.8,-36.57C82.43,-31.59 79.48,-25.4 77.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0011 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>0011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"105\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"105\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0011 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>001&#45;&#45;0011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M93.2,-36.57C95.57,-31.59 98.52,-25.4 100.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M180.8,-72.17C178.43,-67.19 175.48,-61 173.12,-56.05\"/>\n",
       "</g>\n",
       "<!-- 011 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;011 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>01&#45;&#45;011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M189.2,-72.17C191.57,-67.19 194.52,-61 196.88,-56.05\"/>\n",
       "</g>\n",
       "<!-- 0100 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>0100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"137\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"137\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0100 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>010&#45;&#45;0100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M162.52,-38.6C156.97,-32.77 149.02,-24.42 143.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0101 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>0101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0101 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>010&#45;&#45;0101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M169,-35.14C169,-30.97 169,-26.16 169,-22\"/>\n",
       "</g>\n",
       "<!-- 0110 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>0110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 011&#45;&#45;0110 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>011&#45;&#45;0110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-35.14C201,-30.97 201,-26.16 201,-22\"/>\n",
       "</g>\n",
       "<!-- 0111 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>0111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"233\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"233\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 011&#45;&#45;0111 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>011&#45;&#45;0111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M207.48,-38.6C213.03,-32.77 220.98,-24.42 226.53,-18.59\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"313\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"313\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M313,-106.34C313,-102.17 313,-97.36 313,-93.2\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"409\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"409\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M321.52,-113.62C339.69,-107.26 382.08,-92.42 400.36,-86.02\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M308.8,-72.17C306.43,-67.19 303.48,-61 301.12,-56.05\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M317.2,-72.17C319.57,-67.19 322.52,-61 324.88,-56.05\"/>\n",
       "</g>\n",
       "<!-- 1000 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>1000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1000 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>100&#45;&#45;1000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M290.52,-38.6C284.97,-32.77 277.02,-24.42 271.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 1001 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>1001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1001 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>100&#45;&#45;1001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M297,-35.14C297,-30.97 297,-26.16 297,-22\"/>\n",
       "</g>\n",
       "<!-- 1010 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>1010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1010 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>101&#45;&#45;1010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M329,-35.14C329,-30.97 329,-26.16 329,-22\"/>\n",
       "</g>\n",
       "<!-- 1011 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>1011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"361\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"361\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1011 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>101&#45;&#45;1011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M335.48,-38.6C341.03,-32.77 348.98,-24.42 354.53,-18.59\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"409\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"409\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M409,-70.74C409,-66.57 409,-61.76 409,-57.6\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M416.5,-75.75C425.46,-69.47 440.38,-59.03 449.39,-52.72\"/>\n",
       "</g>\n",
       "<!-- 1100 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>1100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"393\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"393\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1100 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>110&#45;&#45;1100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M404.8,-36.57C402.43,-31.59 399.48,-25.4 397.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 1101 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>1101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1101 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>110&#45;&#45;1101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M413.2,-36.57C415.57,-31.59 418.52,-25.4 420.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 1110 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>1110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1110 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>111&#45;&#45;1110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-35.14C457,-30.97 457,-26.16 457,-22\"/>\n",
       "</g>\n",
       "<!-- 1111 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>1111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"489\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"489\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1111 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>111&#45;&#45;1111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M463.48,-38.6C469.03,-32.77 476.98,-24.42 482.53,-18.59\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107df68d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "root_2 = build_bin_tree(list2)\n",
    "print(minimax(root_2, False))\n",
    "root_2.draw_tree(False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(8, [1, 0, 1, 0])\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"506pt\" height=\"172pt\"\n",
       " viewBox=\"0.00 0.00 506.00 172.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 168)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-168 502,-168 502,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-153.2\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-149.32\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"185\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"185\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M256.44,-148.6C240.96,-142.1 208.52,-128.48 193.26,-122.07\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"313\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"313\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M272.5,-146.95C281.46,-140.67 296.38,-130.23 305.39,-123.92\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"89\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"89\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M176.48,-113.62C158.31,-107.26 115.92,-92.42 97.64,-86.02\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"185\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"185\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M185,-106.34C185,-102.17 185,-97.36 185,-93.2\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M81.5,-75.75C72.54,-69.47 57.62,-59.03 48.61,-52.72\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"89\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"89\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M89,-70.74C89,-66.57 89,-61.76 89,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0000 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>0000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"9\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"9\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0000 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>000&#45;&#45;0000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.52,-38.6C28.97,-32.77 21.02,-24.42 15.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0001 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>0001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0001 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>000&#45;&#45;0001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M41,-35.14C41,-30.97 41,-26.16 41,-22\"/>\n",
       "</g>\n",
       "<!-- 0010 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>0010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"73\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"73\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0010 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>001&#45;&#45;0010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M84.8,-36.57C82.43,-31.59 79.48,-25.4 77.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0011 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>0011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"105\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"105\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0011 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>001&#45;&#45;0011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M93.2,-36.57C95.57,-31.59 98.52,-25.4 100.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M180.8,-72.17C178.43,-67.19 175.48,-61 173.12,-56.05\"/>\n",
       "</g>\n",
       "<!-- 011 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;011 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>01&#45;&#45;011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M189.2,-72.17C191.57,-67.19 194.52,-61 196.88,-56.05\"/>\n",
       "</g>\n",
       "<!-- 0100 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>0100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"137\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"137\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0100 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>010&#45;&#45;0100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M162.52,-38.6C156.97,-32.77 149.02,-24.42 143.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0101 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>0101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0101 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>010&#45;&#45;0101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M169,-35.14C169,-30.97 169,-26.16 169,-22\"/>\n",
       "</g>\n",
       "<!-- 0110 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>0110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 011&#45;&#45;0110 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>011&#45;&#45;0110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-35.14C201,-30.97 201,-26.16 201,-22\"/>\n",
       "</g>\n",
       "<!-- 0111 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>0111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"233\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"233\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 011&#45;&#45;0111 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>011&#45;&#45;0111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M207.48,-38.6C213.03,-32.77 220.98,-24.42 226.53,-18.59\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"313\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"313\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M313,-106.34C313,-102.17 313,-97.36 313,-93.2\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"409\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"409\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M321.52,-113.62C339.69,-107.26 382.08,-92.42 400.36,-86.02\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M308.8,-72.17C306.43,-67.19 303.48,-61 301.12,-56.05\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M317.2,-72.17C319.57,-67.19 322.52,-61 324.88,-56.05\"/>\n",
       "</g>\n",
       "<!-- 1000 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>1000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1000 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>100&#45;&#45;1000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M290.52,-38.6C284.97,-32.77 277.02,-24.42 271.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 1001 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>1001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1001 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>100&#45;&#45;1001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M297,-35.14C297,-30.97 297,-26.16 297,-22\"/>\n",
       "</g>\n",
       "<!-- 1010 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>1010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1010 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>101&#45;&#45;1010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M329,-35.14C329,-30.97 329,-26.16 329,-22\"/>\n",
       "</g>\n",
       "<!-- 1011 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>1011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"361\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"361\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1011 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>101&#45;&#45;1011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M335.48,-38.6C341.03,-32.77 348.98,-24.42 354.53,-18.59\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"409\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"409\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M409,-70.74C409,-66.57 409,-61.76 409,-57.6\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M416.5,-75.75C425.46,-69.47 440.38,-59.03 449.39,-52.72\"/>\n",
       "</g>\n",
       "<!-- 1100 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>1100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"393\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"393\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1100 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>110&#45;&#45;1100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M404.8,-36.57C402.43,-31.59 399.48,-25.4 397.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 1101 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>1101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1101 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>110&#45;&#45;1101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M413.2,-36.57C415.57,-31.59 418.52,-25.4 420.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 1110 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>1110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1110 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>111&#45;&#45;1110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-35.14C457,-30.97 457,-26.16 457,-22\"/>\n",
       "</g>\n",
       "<!-- 1111 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>1111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"489\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"489\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1111 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>111&#45;&#45;1111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M463.48,-38.6C469.03,-32.77 476.98,-24.42 482.53,-18.59\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107df7cd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "root_2 = build_bin_tree(list2)\n",
    "print(alpha_beta_pruning(root_2, True))\n",
    "root_2.draw_tree(True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(5, [1, 0, 0, 1])\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"506pt\" height=\"172pt\"\n",
       " viewBox=\"0.00 0.00 506.00 172.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 168)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-168 502,-168 502,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-153.2\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-149.32\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"185\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"185\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M256.44,-148.6C240.96,-142.1 208.52,-128.48 193.26,-122.07\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"313\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"313\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M272.5,-146.95C281.46,-140.67 296.38,-130.23 305.39,-123.92\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"89\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"89\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M176.48,-113.62C158.31,-107.26 115.92,-92.42 97.64,-86.02\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"185\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"185\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M185,-106.34C185,-102.17 185,-97.36 185,-93.2\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M81.5,-75.75C72.54,-69.47 57.62,-59.03 48.61,-52.72\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"89\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"89\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M89,-70.74C89,-66.57 89,-61.76 89,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0000 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>0000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"9\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"9\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0000 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>000&#45;&#45;0000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.52,-38.6C28.97,-32.77 21.02,-24.42 15.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0001 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>0001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0001 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>000&#45;&#45;0001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M41,-35.14C41,-30.97 41,-26.16 41,-22\"/>\n",
       "</g>\n",
       "<!-- 0010 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>0010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"73\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"73\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0010 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>001&#45;&#45;0010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M84.8,-36.57C82.43,-31.59 79.48,-25.4 77.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0011 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>0011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"105\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"105\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0011 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>001&#45;&#45;0011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M93.2,-36.57C95.57,-31.59 98.52,-25.4 100.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M180.8,-72.17C178.43,-67.19 175.48,-61 173.12,-56.05\"/>\n",
       "</g>\n",
       "<!-- 011 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;011 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>01&#45;&#45;011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M189.2,-72.17C191.57,-67.19 194.52,-61 196.88,-56.05\"/>\n",
       "</g>\n",
       "<!-- 0100 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>0100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"137\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"137\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0100 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>010&#45;&#45;0100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M162.52,-38.6C156.97,-32.77 149.02,-24.42 143.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0101 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>0101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0101 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>010&#45;&#45;0101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M169,-35.14C169,-30.97 169,-26.16 169,-22\"/>\n",
       "</g>\n",
       "<!-- 0110 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>0110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 011&#45;&#45;0110 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>011&#45;&#45;0110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-35.14C201,-30.97 201,-26.16 201,-22\"/>\n",
       "</g>\n",
       "<!-- 0111 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>0111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"233\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"233\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 011&#45;&#45;0111 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>011&#45;&#45;0111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M207.48,-38.6C213.03,-32.77 220.98,-24.42 226.53,-18.59\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"313\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"313\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M313,-106.34C313,-102.17 313,-97.36 313,-93.2\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"409\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"409\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M321.52,-113.62C339.69,-107.26 382.08,-92.42 400.36,-86.02\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M308.8,-72.17C306.43,-67.19 303.48,-61 301.12,-56.05\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M317.2,-72.17C319.57,-67.19 322.52,-61 324.88,-56.05\"/>\n",
       "</g>\n",
       "<!-- 1000 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>1000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1000 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>100&#45;&#45;1000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M290.52,-38.6C284.97,-32.77 277.02,-24.42 271.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 1001 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>1001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1001 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>100&#45;&#45;1001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M297,-35.14C297,-30.97 297,-26.16 297,-22\"/>\n",
       "</g>\n",
       "<!-- 1010 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>1010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1010 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>101&#45;&#45;1010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M329,-35.14C329,-30.97 329,-26.16 329,-22\"/>\n",
       "</g>\n",
       "<!-- 1011 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>1011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"361\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"361\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1011 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>101&#45;&#45;1011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M335.48,-38.6C341.03,-32.77 348.98,-24.42 354.53,-18.59\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"409\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"409\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M409,-70.74C409,-66.57 409,-61.76 409,-57.6\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M416.5,-75.75C425.46,-69.47 440.38,-59.03 449.39,-52.72\"/>\n",
       "</g>\n",
       "<!-- 1100 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>1100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"393\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"393\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1100 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>110&#45;&#45;1100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M404.8,-36.57C402.43,-31.59 399.48,-25.4 397.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 1101 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>1101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1101 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>110&#45;&#45;1101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M413.2,-36.57C415.57,-31.59 418.52,-25.4 420.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 1110 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>1110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1110 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>111&#45;&#45;1110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-35.14C457,-30.97 457,-26.16 457,-22\"/>\n",
       "</g>\n",
       "<!-- 1111 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>1111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"489\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"489\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1111 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>111&#45;&#45;1111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M463.48,-38.6C469.03,-32.77 476.98,-24.42 482.53,-18.59\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107dfd0d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "root_2 = build_bin_tree(list2)\n",
    "print(alpha_beta_pruning(root_2, False))\n",
    "root_2.draw_tree(False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4. Text output\n",
    "\n",
    "Old code used when the graphic tree output was not available."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(8, [0, 0, 1])\n",
      "\n",
      "[] 8\n",
      "[0] 8\n",
      "[00] 8\n",
      "[000] 4\n",
      "[001] 8\n",
      "[01] 9\n",
      "[010] 9\n",
      "[011] 3\n",
      "[1] 2\n",
      "[10] 2\n",
      "[100] 2\n",
      "[101] -2\n",
      "[11] 9\n",
      "[110] 9\n",
      "[111] -1\n",
      "\n",
      "[] 8\n",
      "[0] 8\n",
      "[00] 8\n",
      "[000] 4\n",
      "[001] 8\n",
      "[01] 9\n",
      "[010] 9\n",
      "[011] 3\n",
      "[1] 2\n",
      "[10] 2\n",
      "[100] 2\n",
      "[101] -2\n",
      "[11] 9\n",
      "[110] 9\n",
      "[111] -1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# maximizer plays first\n",
    "root = build_bin_tree(list1)\n",
    "max_depth = 3\n",
    "ret_tuple = minimax(root, True)\n",
    "pprint(ret_tuple) # shows the value of the maximized end node and the path to it\n",
    "print()\n",
    "print(root) # all nodes are evaluated and labeled with their values\n",
    "print(root.build_str())\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[] None\n",
      "[0] None\n",
      "[00] None\n",
      "[000] 4\n",
      "[001] 8\n",
      "[01] None\n",
      "[010] 9\n",
      "[011] 3\n",
      "[1] None\n",
      "[10] None\n",
      "[100] 2\n",
      "[101] -2\n",
      "[11] None\n",
      "[110] 9\n",
      "[111] -1\n",
      "\n",
      "(-1, [1, 1, 1])\n",
      "\n",
      "[] -1\n",
      "[0] 4\n",
      "[00] 4\n",
      "[000] 4\n",
      "[001] 8\n",
      "[01] 3\n",
      "[010] 9\n",
      "[011] 3\n",
      "[1] -1\n",
      "[10] -2\n",
      "[100] 2\n",
      "[101] -2\n",
      "[11] -1\n",
      "[110] 9\n",
      "[111] -1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# minimizer plays first\n",
    "node = build_bin_tree([4, 8, 9, 3, 2, -2, 9, -1])\n",
    "print(node) # same as before\n",
    "ret_tuple = minimax(node, False) \n",
    "pprint(ret_tuple) # different from before\n",
    "print()\n",
    "print(node) # different from before"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[] None\n",
      "[0] None\n",
      "[00] None\n",
      "[000] None\n",
      "[0000] 8\n",
      "[0001] 7\n",
      "[001] None\n",
      "[0010] 3\n",
      "[0011] 9\n",
      "[01] None\n",
      "[010] None\n",
      "[0100] 9\n",
      "[0101] 8\n",
      "[011] None\n",
      "[0110] 2\n",
      "[0111] 4\n",
      "[1] None\n",
      "[10] None\n",
      "[100] None\n",
      "[1000] 1\n",
      "[1001] 5\n",
      "[101] None\n",
      "[1010] 8\n",
      "[1011] 9\n",
      "[11] None\n",
      "[110] None\n",
      "[1100] 9\n",
      "[1101] 9\n",
      "[111] None\n",
      "[1110] 3\n",
      "[1111] 4\n",
      "\n"
     ]
    }
   ],
   "source": [
    "list2 = [8, 7, 3, 9, 9, 8, 2, 4, 1, 5, 8, 9, 9, 9, 3, 4]\n",
    "root_2 = build_bin_tree(list2)\n",
    "print(root_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(8, [1, 0, 1, 0])\n",
      "\n",
      "[] 8\n",
      "[0] 7\n",
      "[00] 7\n",
      "[000] 7\n",
      "[0000] 8\n",
      "[0001] 7\n",
      "[001] 3\n",
      "[0010] 3\n",
      "[0011] 9\n",
      "[01] 8\n",
      "[010] 8\n",
      "[0100] 9\n",
      "[0101] 8\n",
      "[011] 2\n",
      "[0110] 2\n",
      "[0111] 4\n",
      "[1] 8\n",
      "[10] 8\n",
      "[100] 1\n",
      "[1000] 1\n",
      "[1001] 5\n",
      "[101] 8\n",
      "[1010] 8\n",
      "[1011] 9\n",
      "[11] 9\n",
      "[110] 9\n",
      "[1100] 9\n",
      "[1101] 9\n",
      "[111] 3\n",
      "[1110] 3\n",
      "[1111] 4\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# maximizer plays first\n",
    "pprint(minimax(root_2, True))\n",
    "print()\n",
    "print(root_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[] None\n",
      "[0] None\n",
      "[00] None\n",
      "[000] None\n",
      "[0000] 8\n",
      "[0001] 7\n",
      "[001] None\n",
      "[0010] 3\n",
      "[0011] 9\n",
      "[01] None\n",
      "[010] None\n",
      "[0100] 9\n",
      "[0101] 8\n",
      "[011] None\n",
      "[0110] 2\n",
      "[0111] 4\n",
      "[1] None\n",
      "[10] None\n",
      "[100] None\n",
      "[1000] 1\n",
      "[1001] 5\n",
      "[101] None\n",
      "[1010] 8\n",
      "[1011] 9\n",
      "[11] None\n",
      "[110] None\n",
      "[1100] 9\n",
      "[1101] 9\n",
      "[111] None\n",
      "[1110] 3\n",
      "[1111] 4\n",
      "\n",
      "(5, [1, 0, 0, 1])\n",
      "\n",
      "[] 5\n",
      "[0] 8\n",
      "[00] 8\n",
      "[000] 8\n",
      "[0000] 8\n",
      "[0001] 7\n",
      "[001] 9\n",
      "[0010] 3\n",
      "[0011] 9\n",
      "[01] 4\n",
      "[010] 9\n",
      "[0100] 9\n",
      "[0101] 8\n",
      "[011] 4\n",
      "[0110] 2\n",
      "[0111] 4\n",
      "[1] 5\n",
      "[10] 5\n",
      "[100] 5\n",
      "[1000] 1\n",
      "[1001] 5\n",
      "[101] 9\n",
      "[1010] 8\n",
      "[1011] 9\n",
      "[11] 4\n",
      "[110] 9\n",
      "[1100] 9\n",
      "[1101] 9\n",
      "[111] 4\n",
      "[1110] 3\n",
      "[1111] 4\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# minimizer plays first\n",
    "root_2 = build_bin_tree(list2)\n",
    "print(root_2)\n",
    "pprint(minimax(root_2, False))\n",
    "print()\n",
    "print(root_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(8, [0, 0, 1])\n",
      "\n",
      "[] 8\n",
      "[0] 8\n",
      "[00] 8\n",
      "[000] 4\n",
      "[001] 8\n",
      "[01] 9\n",
      "[010] 9\n",
      "[011] 3\n",
      "[1] 2\n",
      "[10] 2\n",
      "[100] 2\n",
      "[101] -2\n",
      "[11] None\n",
      "[110] 9\n",
      "[111] -1\n",
      "\n",
      "[] 8\n",
      "[0] 8\n",
      "[00] 8\n",
      "[000] 4\n",
      "[001] 8\n",
      "[01] 9\n",
      "[010] 9\n",
      "[011] unevaluated\n",
      "[1] 2\n",
      "[10] 2\n",
      "[100] 2\n",
      "[101] -2\n",
      "[11] unevaluated\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Now, alpha-beta pruning.\n",
    "# Maximizer plays first.\n",
    "\n",
    "root = build_bin_tree(list1)\n",
    "max_depth = 3\n",
    "ret_tuple = alpha_beta_pruning(root, True)\n",
    "pprint(ret_tuple)\n",
    "print()\n",
    "print(root)\n",
    "print(root.build_str()) # shows the effect of pruning clearly"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(-1, [1, 1, 1])\n",
      "\n",
      "[] -1\n",
      "[0] 4\n",
      "[00] 4\n",
      "[000] 4\n",
      "[001] 8\n",
      "[01] 3\n",
      "[010] 9\n",
      "[011] 3\n",
      "[1] -1\n",
      "[10] -2\n",
      "[100] 2\n",
      "[101] -2\n",
      "[11] -1\n",
      "[110] 9\n",
      "[111] -1\n",
      "\n",
      "[] -1\n",
      "[0] 4\n",
      "[00] 4\n",
      "[000] 4\n",
      "[001] 8\n",
      "[01] 3\n",
      "[010] 9\n",
      "[011] 3\n",
      "[1] -1\n",
      "[10] -2\n",
      "[100] 2\n",
      "[101] -2\n",
      "[11] -1\n",
      "[110] 9\n",
      "[111] -1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Now, alpha-beta pruning.\n",
    "# Minimizer plays first.\n",
    "\n",
    "root = build_bin_tree(list1)\n",
    "max_depth = 3\n",
    "ret_tuple = alpha_beta_pruning(root, False)\n",
    "pprint(ret_tuple)\n",
    "print()\n",
    "print(root)\n",
    "print(root.build_str()) # shows the effect of pruning clearly"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(8, [1, 0, 1, 0])\n",
      "\n",
      "[] 8\n",
      "[0] 7\n",
      "[00] 7\n",
      "[000] 7\n",
      "[0000] 8\n",
      "[0001] 7\n",
      "[001] 3\n",
      "[0010] 3\n",
      "[0011] 9\n",
      "[01] 8\n",
      "[010] 8\n",
      "[0100] 9\n",
      "[0101] 8\n",
      "[011] None\n",
      "[0110] 2\n",
      "[0111] 4\n",
      "[1] 8\n",
      "[10] 8\n",
      "[100] 1\n",
      "[1000] 1\n",
      "[1001] 5\n",
      "[101] 8\n",
      "[1010] 8\n",
      "[1011] 9\n",
      "[11] 9\n",
      "[110] 9\n",
      "[1100] 9\n",
      "[1101] 9\n",
      "[111] None\n",
      "[1110] 3\n",
      "[1111] 4\n",
      "\n",
      "[] 8\n",
      "[0] 7\n",
      "[00] 7\n",
      "[000] 7\n",
      "[0000] 8\n",
      "[0001] 7\n",
      "[001] 3\n",
      "[0010] 3\n",
      "[0011] unevaluated\n",
      "[01] 8\n",
      "[010] 8\n",
      "[0100] 9\n",
      "[0101] 8\n",
      "[011] unevaluated\n",
      "[1] 8\n",
      "[10] 8\n",
      "[100] 1\n",
      "[1000] 1\n",
      "[1001] unevaluated\n",
      "[101] 8\n",
      "[1010] 8\n",
      "[1011] 9\n",
      "[11] 9\n",
      "[110] 9\n",
      "[1100] 9\n",
      "[1101] 9\n",
      "[111] unevaluated\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Now, alpha-beta pruning.\n",
    "# Maximizer plays first.\n",
    "root_2 = build_bin_tree(list2)\n",
    "pprint(alpha_beta_pruning(root_2,  True))\n",
    "print()\n",
    "print(root_2)\n",
    "print(root_2.build_str()) # shows the effect of pruning clearly"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(5, [1, 0, 0, 1])\n",
      "\n",
      "[] 5\n",
      "[0] 8\n",
      "[00] 8\n",
      "[000] 8\n",
      "[0000] 8\n",
      "[0001] 7\n",
      "[001] 9\n",
      "[0010] 3\n",
      "[0011] 9\n",
      "[01] 4\n",
      "[010] 9\n",
      "[0100] 9\n",
      "[0101] 8\n",
      "[011] 4\n",
      "[0110] 2\n",
      "[0111] 4\n",
      "[1] 5\n",
      "[10] 5\n",
      "[100] 5\n",
      "[1000] 1\n",
      "[1001] 5\n",
      "[101] 8\n",
      "[1010] 8\n",
      "[1011] 9\n",
      "[11] 4\n",
      "[110] 9\n",
      "[1100] 9\n",
      "[1101] 9\n",
      "[111] 4\n",
      "[1110] 3\n",
      "[1111] 4\n",
      "\n",
      "[] 5\n",
      "[0] 8\n",
      "[00] 8\n",
      "[000] 8\n",
      "[0000] 8\n",
      "[0001] 7\n",
      "[001] 9\n",
      "[0010] 3\n",
      "[0011] 9\n",
      "[01] 4\n",
      "[010] 9\n",
      "[0100] 9\n",
      "[0101] 8\n",
      "[011] 4\n",
      "[0110] 2\n",
      "[0111] 4\n",
      "[1] 5\n",
      "[10] 5\n",
      "[100] 5\n",
      "[1000] 1\n",
      "[1001] 5\n",
      "[101] 8\n",
      "[1010] 8\n",
      "[1011] unevaluated\n",
      "[11] 4\n",
      "[110] 9\n",
      "[1100] 9\n",
      "[1101] unevaluated\n",
      "[111] 4\n",
      "[1110] 3\n",
      "[1111] 4\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Now, alpha-beta pruning.\n",
    "# Maximizer plays first.\n",
    "root_2 = build_bin_tree(list2)\n",
    "pprint(alpha_beta_pruning(root_2, False))\n",
    "print()\n",
    "print(root_2)\n",
    "print(root_2.build_str()) # shows the effect of pruning clearly"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5. Non-binary tree example\n",
    "\n",
    "The tree structure is fixed.  Only the values of the terminals can vary."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"506pt\" height=\"172pt\"\n",
       " viewBox=\"0.00 0.00 506.00 172.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 168)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-168 502,-168 502,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"317\" cy=\"-153.2\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"317\" y=\"-149.32\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M308.28,-149.67C286.78,-143.45 231.42,-127.41 209.82,-121.15\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"317\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"317\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M317,-141.94C317,-137.77 317,-132.96 317,-128.8\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>2</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;2 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>root&#45;&#45;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M325.84,-149.45C346.19,-143.12 396.16,-127.57 416.34,-121.29\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"121\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M192.44,-113C176.96,-106.5 144.52,-92.88 129.26,-86.47\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-106.34C201,-102.17 201,-97.36 201,-93.2\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M112.44,-77.4C96.96,-70.9 64.52,-57.28 49.26,-50.87\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"121\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M121,-70.74C121,-66.57 121,-61.76 121,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0000 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>0000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"9\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"9\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0000 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>000&#45;&#45;0000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.52,-38.6C28.97,-32.77 21.02,-24.42 15.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0001 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>0001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0001 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>000&#45;&#45;0001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M41,-35.14C41,-30.97 41,-26.16 41,-22\"/>\n",
       "</g>\n",
       "<!-- 0010 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>0010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"73\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"73\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0010 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>001&#45;&#45;0010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M113.5,-40.15C104.54,-33.87 89.62,-23.43 80.61,-17.12\"/>\n",
       "</g>\n",
       "<!-- 0011 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>0011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"105\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"105\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0011 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>001&#45;&#45;0011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M116.8,-36.57C114.43,-31.59 111.48,-25.4 109.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0012 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>0012</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"137\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"137\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0012 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>001&#45;&#45;0012</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M125.2,-36.57C127.57,-31.59 130.52,-25.4 132.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0013 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>0013</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0013 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>001&#45;&#45;0013</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M128.5,-40.15C137.46,-33.87 152.38,-23.43 161.39,-17.12\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-70.74C201,-66.57 201,-61.76 201,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0100 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>0100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0100 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>010&#45;&#45;0100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-35.14C201,-30.97 201,-26.16 201,-22\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"293\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"293\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M311.43,-108.81C307.56,-103.38 302.41,-96.17 298.54,-90.75\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"333\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"333\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M321.2,-107.77C323.57,-102.79 326.52,-96.6 328.88,-91.65\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"245\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"245\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M285.5,-75.75C276.54,-69.47 261.62,-59.03 252.61,-52.72\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"293\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"293\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M293,-70.74C293,-66.57 293,-61.76 293,-57.6\"/>\n",
       "</g>\n",
       "<!-- 1000 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>1000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"233\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"233\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1000 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>100&#45;&#45;1000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M241.59,-35.86C239.93,-31.21 237.94,-25.63 236.29,-21.01\"/>\n",
       "</g>\n",
       "<!-- 1010 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>1010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1010 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>101&#45;&#45;1010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M286.79,-37.94C282.15,-32.37 275.83,-24.8 271.2,-19.24\"/>\n",
       "</g>\n",
       "<!-- 1011 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>1011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1011 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>101&#45;&#45;1011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M294.18,-35.5C294.69,-31.16 295.3,-26.09 295.82,-21.74\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"333\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"333\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M333,-70.74C333,-66.57 333,-61.76 333,-57.6\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"381\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"381\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M340.5,-75.75C349.46,-69.47 364.38,-59.03 373.39,-52.72\"/>\n",
       "</g>\n",
       "<!-- 1100 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>1100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1100 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>110&#45;&#45;1100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M331.82,-35.5C331.31,-31.16 330.7,-26.09 330.18,-21.74\"/>\n",
       "</g>\n",
       "<!-- 1101 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>1101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"361\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"361\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1101 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>110&#45;&#45;1101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M339.21,-37.94C343.85,-32.37 350.17,-24.8 354.8,-19.24\"/>\n",
       "</g>\n",
       "<!-- 1110 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>1110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"393\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"393\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1110 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>111&#45;&#45;1110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M384.41,-35.86C386.07,-31.21 388.06,-25.63 389.71,-21.01\"/>\n",
       "</g>\n",
       "<!-- 20 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>20</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 2&#45;&#45;20 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>2&#45;&#45;20</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-106.34C425,-102.17 425,-97.36 425,-93.2\"/>\n",
       "</g>\n",
       "<!-- 21 -->\n",
       "<g id=\"node32\" class=\"node\">\n",
       "<title>21</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">max</text>\n",
       "</g>\n",
       "<!-- 2&#45;&#45;21 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>2&#45;&#45;21</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M431.48,-109.8C437.03,-103.97 444.98,-95.62 450.53,-89.79\"/>\n",
       "</g>\n",
       "<!-- 200 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>200</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 20&#45;&#45;200 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>20&#45;&#45;200</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-70.74C425,-66.57 425,-61.76 425,-57.6\"/>\n",
       "</g>\n",
       "<!-- 2000 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>2000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 200&#45;&#45;2000 -->\n",
       "<g id=\"edge31\" class=\"edge\">\n",
       "<title>200&#45;&#45;2000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-35.14C425,-30.97 425,-26.16 425,-22\"/>\n",
       "</g>\n",
       "<!-- 210 -->\n",
       "<g id=\"node33\" class=\"node\">\n",
       "<title>210</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">min</text>\n",
       "</g>\n",
       "<!-- 21&#45;&#45;210 -->\n",
       "<g id=\"edge32\" class=\"edge\">\n",
       "<title>21&#45;&#45;210</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-70.74C457,-66.57 457,-61.76 457,-57.6\"/>\n",
       "</g>\n",
       "<!-- 2100 -->\n",
       "<g id=\"node34\" class=\"node\">\n",
       "<title>2100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">11</text>\n",
       "</g>\n",
       "<!-- 210&#45;&#45;2100 -->\n",
       "<g id=\"edge33\" class=\"edge\">\n",
       "<title>210&#45;&#45;2100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-35.14C457,-30.97 457,-26.16 457,-22\"/>\n",
       "</g>\n",
       "<!-- 2101 -->\n",
       "<g id=\"node35\" class=\"node\">\n",
       "<title>2101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"489\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"489\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 210&#45;&#45;2101 -->\n",
       "<g id=\"edge34\" class=\"edge\">\n",
       "<title>210&#45;&#45;2101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M463.48,-38.6C469.03,-32.77 476.98,-24.42 482.53,-18.59\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107dd6990>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "num_li = [random.randint(1, 11) for _ in range(16)]\n",
    "root = build_my_tree(num_li)\n",
    "root.draw_tree()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(6, [1, 1, 1, 0])\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"506pt\" height=\"172pt\"\n",
       " viewBox=\"0.00 0.00 506.00 172.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 168)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-168 502,-168 502,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"317\" cy=\"-153.2\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"317\" y=\"-149.32\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M308.28,-149.67C286.78,-143.45 231.42,-127.41 209.82,-121.15\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"317\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"317\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M317,-141.94C317,-137.77 317,-132.96 317,-128.8\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>2</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;2 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>root&#45;&#45;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M325.84,-149.45C346.19,-143.12 396.16,-127.57 416.34,-121.29\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"121\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M192.44,-113C176.96,-106.5 144.52,-92.88 129.26,-86.47\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-106.34C201,-102.17 201,-97.36 201,-93.2\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M112.44,-77.4C96.96,-70.9 64.52,-57.28 49.26,-50.87\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"121\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M121,-70.74C121,-66.57 121,-61.76 121,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0000 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>0000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"9\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"9\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0000 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>000&#45;&#45;0000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.52,-38.6C28.97,-32.77 21.02,-24.42 15.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0001 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>0001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0001 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>000&#45;&#45;0001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M41,-35.14C41,-30.97 41,-26.16 41,-22\"/>\n",
       "</g>\n",
       "<!-- 0010 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>0010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"73\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"73\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0010 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>001&#45;&#45;0010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M113.5,-40.15C104.54,-33.87 89.62,-23.43 80.61,-17.12\"/>\n",
       "</g>\n",
       "<!-- 0011 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>0011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"105\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"105\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0011 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>001&#45;&#45;0011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M116.8,-36.57C114.43,-31.59 111.48,-25.4 109.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0012 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>0012</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"137\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"137\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0012 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>001&#45;&#45;0012</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M125.2,-36.57C127.57,-31.59 130.52,-25.4 132.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0013 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>0013</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0013 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>001&#45;&#45;0013</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M128.5,-40.15C137.46,-33.87 152.38,-23.43 161.39,-17.12\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-70.74C201,-66.57 201,-61.76 201,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0100 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>0100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0100 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>010&#45;&#45;0100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-35.14C201,-30.97 201,-26.16 201,-22\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"293\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"293\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M311.43,-108.81C307.56,-103.38 302.41,-96.17 298.54,-90.75\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"333\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"333\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M321.2,-107.77C323.57,-102.79 326.52,-96.6 328.88,-91.65\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"245\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"245\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M285.5,-75.75C276.54,-69.47 261.62,-59.03 252.61,-52.72\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"293\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"293\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M293,-70.74C293,-66.57 293,-61.76 293,-57.6\"/>\n",
       "</g>\n",
       "<!-- 1000 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>1000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"233\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"233\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1000 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>100&#45;&#45;1000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M241.59,-35.86C239.93,-31.21 237.94,-25.63 236.29,-21.01\"/>\n",
       "</g>\n",
       "<!-- 1010 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>1010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1010 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>101&#45;&#45;1010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M286.79,-37.94C282.15,-32.37 275.83,-24.8 271.2,-19.24\"/>\n",
       "</g>\n",
       "<!-- 1011 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>1011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1011 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>101&#45;&#45;1011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M294.18,-35.5C294.69,-31.16 295.3,-26.09 295.82,-21.74\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"333\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"333\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M333,-70.74C333,-66.57 333,-61.76 333,-57.6\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"381\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"381\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M340.5,-75.75C349.46,-69.47 364.38,-59.03 373.39,-52.72\"/>\n",
       "</g>\n",
       "<!-- 1100 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>1100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1100 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>110&#45;&#45;1100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M331.82,-35.5C331.31,-31.16 330.7,-26.09 330.18,-21.74\"/>\n",
       "</g>\n",
       "<!-- 1101 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>1101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"361\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"361\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1101 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>110&#45;&#45;1101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M339.21,-37.94C343.85,-32.37 350.17,-24.8 354.8,-19.24\"/>\n",
       "</g>\n",
       "<!-- 1110 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>1110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"393\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"393\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1110 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>111&#45;&#45;1110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M384.41,-35.86C386.07,-31.21 388.06,-25.63 389.71,-21.01\"/>\n",
       "</g>\n",
       "<!-- 20 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>20</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 2&#45;&#45;20 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>2&#45;&#45;20</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-106.34C425,-102.17 425,-97.36 425,-93.2\"/>\n",
       "</g>\n",
       "<!-- 21 -->\n",
       "<g id=\"node32\" class=\"node\">\n",
       "<title>21</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 2&#45;&#45;21 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>2&#45;&#45;21</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M431.48,-109.8C437.03,-103.97 444.98,-95.62 450.53,-89.79\"/>\n",
       "</g>\n",
       "<!-- 200 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>200</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 20&#45;&#45;200 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>20&#45;&#45;200</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-70.74C425,-66.57 425,-61.76 425,-57.6\"/>\n",
       "</g>\n",
       "<!-- 2000 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>2000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 200&#45;&#45;2000 -->\n",
       "<g id=\"edge31\" class=\"edge\">\n",
       "<title>200&#45;&#45;2000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-35.14C425,-30.97 425,-26.16 425,-22\"/>\n",
       "</g>\n",
       "<!-- 210 -->\n",
       "<g id=\"node33\" class=\"node\">\n",
       "<title>210</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 21&#45;&#45;210 -->\n",
       "<g id=\"edge32\" class=\"edge\">\n",
       "<title>21&#45;&#45;210</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-70.74C457,-66.57 457,-61.76 457,-57.6\"/>\n",
       "</g>\n",
       "<!-- 2100 -->\n",
       "<g id=\"node34\" class=\"node\">\n",
       "<title>2100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">11</text>\n",
       "</g>\n",
       "<!-- 210&#45;&#45;2100 -->\n",
       "<g id=\"edge33\" class=\"edge\">\n",
       "<title>210&#45;&#45;2100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-35.14C457,-30.97 457,-26.16 457,-22\"/>\n",
       "</g>\n",
       "<!-- 2101 -->\n",
       "<g id=\"node35\" class=\"node\">\n",
       "<title>2101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"489\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"489\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 210&#45;&#45;2101 -->\n",
       "<g id=\"edge34\" class=\"edge\">\n",
       "<title>210&#45;&#45;2101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M463.48,-38.6C469.03,-32.77 476.98,-24.42 482.53,-18.59\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107dff490>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(minimax(root, True))\n",
    "root.draw_tree()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(6, [1, 1, 1, 0])\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"506pt\" height=\"172pt\"\n",
       " viewBox=\"0.00 0.00 506.00 172.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 168)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-168 502,-168 502,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"317\" cy=\"-153.2\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"317\" y=\"-149.32\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M308.28,-149.67C286.78,-143.45 231.42,-127.41 209.82,-121.15\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"317\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"317\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M317,-141.94C317,-137.77 317,-132.96 317,-128.8\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>2</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;2 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>root&#45;&#45;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M325.84,-149.45C346.19,-143.12 396.16,-127.57 416.34,-121.29\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"121\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M192.44,-113C176.96,-106.5 144.52,-92.88 129.26,-86.47\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-106.34C201,-102.17 201,-97.36 201,-93.2\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M112.44,-77.4C96.96,-70.9 64.52,-57.28 49.26,-50.87\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"121\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M121,-70.74C121,-66.57 121,-61.76 121,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0000 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>0000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"9\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"9\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0000 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>000&#45;&#45;0000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.52,-38.6C28.97,-32.77 21.02,-24.42 15.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0001 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>0001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0001 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>000&#45;&#45;0001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M41,-35.14C41,-30.97 41,-26.16 41,-22\"/>\n",
       "</g>\n",
       "<!-- 0010 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>0010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"73\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"73\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0010 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>001&#45;&#45;0010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M113.5,-40.15C104.54,-33.87 89.62,-23.43 80.61,-17.12\"/>\n",
       "</g>\n",
       "<!-- 0011 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>0011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"105\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"105\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0011 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>001&#45;&#45;0011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M116.8,-36.57C114.43,-31.59 111.48,-25.4 109.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0012 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>0012</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"137\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"137\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0012 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>001&#45;&#45;0012</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M125.2,-36.57C127.57,-31.59 130.52,-25.4 132.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0013 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>0013</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0013 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>001&#45;&#45;0013</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M128.5,-40.15C137.46,-33.87 152.38,-23.43 161.39,-17.12\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-70.74C201,-66.57 201,-61.76 201,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0100 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>0100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0100 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>010&#45;&#45;0100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-35.14C201,-30.97 201,-26.16 201,-22\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"293\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"293\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M311.43,-108.81C307.56,-103.38 302.41,-96.17 298.54,-90.75\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"333\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"333\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M321.2,-107.77C323.57,-102.79 326.52,-96.6 328.88,-91.65\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"245\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"245\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M285.5,-75.75C276.54,-69.47 261.62,-59.03 252.61,-52.72\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"293\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"293\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M293,-70.74C293,-66.57 293,-61.76 293,-57.6\"/>\n",
       "</g>\n",
       "<!-- 1000 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>1000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"233\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"233\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1000 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>100&#45;&#45;1000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M241.59,-35.86C239.93,-31.21 237.94,-25.63 236.29,-21.01\"/>\n",
       "</g>\n",
       "<!-- 1010 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>1010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1010 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>101&#45;&#45;1010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M286.79,-37.94C282.15,-32.37 275.83,-24.8 271.2,-19.24\"/>\n",
       "</g>\n",
       "<!-- 1011 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>1011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1011 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>101&#45;&#45;1011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M294.18,-35.5C294.69,-31.16 295.3,-26.09 295.82,-21.74\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"333\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"333\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M333,-70.74C333,-66.57 333,-61.76 333,-57.6\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"381\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"381\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M340.5,-75.75C349.46,-69.47 364.38,-59.03 373.39,-52.72\"/>\n",
       "</g>\n",
       "<!-- 1100 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>1100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1100 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>110&#45;&#45;1100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M331.82,-35.5C331.31,-31.16 330.7,-26.09 330.18,-21.74\"/>\n",
       "</g>\n",
       "<!-- 1101 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>1101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"361\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"361\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1101 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>110&#45;&#45;1101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M339.21,-37.94C343.85,-32.37 350.17,-24.8 354.8,-19.24\"/>\n",
       "</g>\n",
       "<!-- 1110 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>1110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"393\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"393\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1110 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>111&#45;&#45;1110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M384.41,-35.86C386.07,-31.21 388.06,-25.63 389.71,-21.01\"/>\n",
       "</g>\n",
       "<!-- 20 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>20</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 2&#45;&#45;20 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>2&#45;&#45;20</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-106.34C425,-102.17 425,-97.36 425,-93.2\"/>\n",
       "</g>\n",
       "<!-- 21 -->\n",
       "<g id=\"node32\" class=\"node\">\n",
       "<title>21</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 2&#45;&#45;21 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>2&#45;&#45;21</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M431.48,-109.8C437.03,-103.97 444.98,-95.62 450.53,-89.79\"/>\n",
       "</g>\n",
       "<!-- 200 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>200</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 20&#45;&#45;200 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>20&#45;&#45;200</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-70.74C425,-66.57 425,-61.76 425,-57.6\"/>\n",
       "</g>\n",
       "<!-- 2000 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>2000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 200&#45;&#45;2000 -->\n",
       "<g id=\"edge31\" class=\"edge\">\n",
       "<title>200&#45;&#45;2000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-35.14C425,-30.97 425,-26.16 425,-22\"/>\n",
       "</g>\n",
       "<!-- 210 -->\n",
       "<g id=\"node33\" class=\"node\">\n",
       "<title>210</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 21&#45;&#45;210 -->\n",
       "<g id=\"edge32\" class=\"edge\">\n",
       "<title>21&#45;&#45;210</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-70.74C457,-66.57 457,-61.76 457,-57.6\"/>\n",
       "</g>\n",
       "<!-- 2100 -->\n",
       "<g id=\"node34\" class=\"node\">\n",
       "<title>2100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 210&#45;&#45;2100 -->\n",
       "<g id=\"edge33\" class=\"edge\">\n",
       "<title>210&#45;&#45;2100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-35.14C457,-30.97 457,-26.16 457,-22\"/>\n",
       "</g>\n",
       "<!-- 2101 -->\n",
       "<g id=\"node35\" class=\"node\">\n",
       "<title>2101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"489\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"489\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 210&#45;&#45;2101 -->\n",
       "<g id=\"edge34\" class=\"edge\">\n",
       "<title>210&#45;&#45;2101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M463.48,-38.6C469.03,-32.77 476.98,-24.42 482.53,-18.59\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107dd7690>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "root = build_my_tree(num_li)\n",
    "print(alpha_beta_pruning(root, True))\n",
    "root.draw_tree()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(8, [0, 0, 0, 1])\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"506pt\" height=\"172pt\"\n",
       " viewBox=\"0.00 0.00 506.00 172.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 168)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-168 502,-168 502,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"317\" cy=\"-153.2\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"317\" y=\"-149.32\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M308.28,-149.67C286.78,-143.45 231.42,-127.41 209.82,-121.15\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"317\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"317\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M317,-141.94C317,-137.77 317,-132.96 317,-128.8\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>2</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;2 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>root&#45;&#45;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M325.84,-149.45C346.19,-143.12 396.16,-127.57 416.34,-121.29\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"121\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M192.44,-113C176.96,-106.5 144.52,-92.88 129.26,-86.47\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-106.34C201,-102.17 201,-97.36 201,-93.2\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M112.44,-77.4C96.96,-70.9 64.52,-57.28 49.26,-50.87\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"121\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M121,-70.74C121,-66.57 121,-61.76 121,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0000 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>0000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"9\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"9\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0000 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>000&#45;&#45;0000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.52,-38.6C28.97,-32.77 21.02,-24.42 15.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0001 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>0001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0001 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>000&#45;&#45;0001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M41,-35.14C41,-30.97 41,-26.16 41,-22\"/>\n",
       "</g>\n",
       "<!-- 0010 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>0010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"73\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"73\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0010 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>001&#45;&#45;0010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M113.5,-40.15C104.54,-33.87 89.62,-23.43 80.61,-17.12\"/>\n",
       "</g>\n",
       "<!-- 0011 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>0011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"105\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"105\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0011 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>001&#45;&#45;0011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M116.8,-36.57C114.43,-31.59 111.48,-25.4 109.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0012 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>0012</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"137\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"137\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0012 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>001&#45;&#45;0012</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M125.2,-36.57C127.57,-31.59 130.52,-25.4 132.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0013 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>0013</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0013 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>001&#45;&#45;0013</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M128.5,-40.15C137.46,-33.87 152.38,-23.43 161.39,-17.12\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-70.74C201,-66.57 201,-61.76 201,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0100 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>0100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0100 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>010&#45;&#45;0100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-35.14C201,-30.97 201,-26.16 201,-22\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"293\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"293\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M311.43,-108.81C307.56,-103.38 302.41,-96.17 298.54,-90.75\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"333\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"333\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M321.2,-107.77C323.57,-102.79 326.52,-96.6 328.88,-91.65\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"245\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"245\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M285.5,-75.75C276.54,-69.47 261.62,-59.03 252.61,-52.72\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"293\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"293\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M293,-70.74C293,-66.57 293,-61.76 293,-57.6\"/>\n",
       "</g>\n",
       "<!-- 1000 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>1000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"233\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"233\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1000 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>100&#45;&#45;1000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M241.59,-35.86C239.93,-31.21 237.94,-25.63 236.29,-21.01\"/>\n",
       "</g>\n",
       "<!-- 1010 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>1010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1010 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>101&#45;&#45;1010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M286.79,-37.94C282.15,-32.37 275.83,-24.8 271.2,-19.24\"/>\n",
       "</g>\n",
       "<!-- 1011 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>1011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1011 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>101&#45;&#45;1011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M294.18,-35.5C294.69,-31.16 295.3,-26.09 295.82,-21.74\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"333\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"333\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M333,-70.74C333,-66.57 333,-61.76 333,-57.6\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"381\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"381\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M340.5,-75.75C349.46,-69.47 364.38,-59.03 373.39,-52.72\"/>\n",
       "</g>\n",
       "<!-- 1100 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>1100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1100 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>110&#45;&#45;1100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M331.82,-35.5C331.31,-31.16 330.7,-26.09 330.18,-21.74\"/>\n",
       "</g>\n",
       "<!-- 1101 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>1101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"361\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"361\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1101 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>110&#45;&#45;1101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M339.21,-37.94C343.85,-32.37 350.17,-24.8 354.8,-19.24\"/>\n",
       "</g>\n",
       "<!-- 1110 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>1110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"393\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"393\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1110 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>111&#45;&#45;1110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M384.41,-35.86C386.07,-31.21 388.06,-25.63 389.71,-21.01\"/>\n",
       "</g>\n",
       "<!-- 20 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>20</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 2&#45;&#45;20 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>2&#45;&#45;20</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-106.34C425,-102.17 425,-97.36 425,-93.2\"/>\n",
       "</g>\n",
       "<!-- 21 -->\n",
       "<g id=\"node32\" class=\"node\">\n",
       "<title>21</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 2&#45;&#45;21 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>2&#45;&#45;21</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M431.48,-109.8C437.03,-103.97 444.98,-95.62 450.53,-89.79\"/>\n",
       "</g>\n",
       "<!-- 200 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>200</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 20&#45;&#45;200 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>20&#45;&#45;200</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-70.74C425,-66.57 425,-61.76 425,-57.6\"/>\n",
       "</g>\n",
       "<!-- 2000 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>2000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 200&#45;&#45;2000 -->\n",
       "<g id=\"edge31\" class=\"edge\">\n",
       "<title>200&#45;&#45;2000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-35.14C425,-30.97 425,-26.16 425,-22\"/>\n",
       "</g>\n",
       "<!-- 210 -->\n",
       "<g id=\"node33\" class=\"node\">\n",
       "<title>210</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 21&#45;&#45;210 -->\n",
       "<g id=\"edge32\" class=\"edge\">\n",
       "<title>21&#45;&#45;210</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-70.74C457,-66.57 457,-61.76 457,-57.6\"/>\n",
       "</g>\n",
       "<!-- 2100 -->\n",
       "<g id=\"node34\" class=\"node\">\n",
       "<title>2100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 210&#45;&#45;2100 -->\n",
       "<g id=\"edge33\" class=\"edge\">\n",
       "<title>210&#45;&#45;2100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-35.14C457,-30.97 457,-26.16 457,-22\"/>\n",
       "</g>\n",
       "<!-- 2101 -->\n",
       "<g id=\"node35\" class=\"node\">\n",
       "<title>2101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"489\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"489\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 210&#45;&#45;2101 -->\n",
       "<g id=\"edge34\" class=\"edge\">\n",
       "<title>210&#45;&#45;2101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M463.48,-38.6C469.03,-32.77 476.98,-24.42 482.53,-18.59\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107df65d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "num_li = [9, 8, 4, 7, 4, 1, 10, 5, 7, 9, 11, 10, 3, 8, 7, 5]\n",
    "root = build_my_tree(num_li)\n",
    "print(alpha_beta_pruning(root, True))\n",
    "root.draw_tree(True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(5, [1, 0, 0, 0])\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"506pt\" height=\"172pt\"\n",
       " viewBox=\"0.00 0.00 506.00 172.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 168)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-168 502,-168 502,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"317\" cy=\"-153.2\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"317\" y=\"-149.32\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M308.28,-149.67C286.78,-143.45 231.42,-127.41 209.82,-121.15\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"317\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"317\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M317,-141.94C317,-137.77 317,-132.96 317,-128.8\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>2</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;2 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>root&#45;&#45;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M325.84,-149.45C346.19,-143.12 396.16,-127.57 416.34,-121.29\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"121\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M192.44,-113C176.96,-106.5 144.52,-92.88 129.26,-86.47\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-106.34C201,-102.17 201,-97.36 201,-93.2\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M112.44,-77.4C96.96,-70.9 64.52,-57.28 49.26,-50.87\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"121\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M121,-70.74C121,-66.57 121,-61.76 121,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0000 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>0000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"9\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"9\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0000 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>000&#45;&#45;0000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.52,-38.6C28.97,-32.77 21.02,-24.42 15.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0001 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>0001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0001 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>000&#45;&#45;0001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M41,-35.14C41,-30.97 41,-26.16 41,-22\"/>\n",
       "</g>\n",
       "<!-- 0010 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>0010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"73\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"73\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0010 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>001&#45;&#45;0010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M113.5,-40.15C104.54,-33.87 89.62,-23.43 80.61,-17.12\"/>\n",
       "</g>\n",
       "<!-- 0011 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>0011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"105\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"105\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0011 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>001&#45;&#45;0011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M116.8,-36.57C114.43,-31.59 111.48,-25.4 109.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0012 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>0012</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"137\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"137\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0012 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>001&#45;&#45;0012</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M125.2,-36.57C127.57,-31.59 130.52,-25.4 132.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0013 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>0013</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0013 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>001&#45;&#45;0013</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M128.5,-40.15C137.46,-33.87 152.38,-23.43 161.39,-17.12\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-70.74C201,-66.57 201,-61.76 201,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0100 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>0100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0100 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>010&#45;&#45;0100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-35.14C201,-30.97 201,-26.16 201,-22\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"293\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"293\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M311.43,-108.81C307.56,-103.38 302.41,-96.17 298.54,-90.75\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"333\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"333\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M321.2,-107.77C323.57,-102.79 326.52,-96.6 328.88,-91.65\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"245\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"245\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M285.5,-75.75C276.54,-69.47 261.62,-59.03 252.61,-52.72\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"293\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"293\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M293,-70.74C293,-66.57 293,-61.76 293,-57.6\"/>\n",
       "</g>\n",
       "<!-- 1000 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>1000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"233\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"233\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1000 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>100&#45;&#45;1000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M241.59,-35.86C239.93,-31.21 237.94,-25.63 236.29,-21.01\"/>\n",
       "</g>\n",
       "<!-- 1010 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>1010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1010 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>101&#45;&#45;1010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M286.79,-37.94C282.15,-32.37 275.83,-24.8 271.2,-19.24\"/>\n",
       "</g>\n",
       "<!-- 1011 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>1011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1011 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>101&#45;&#45;1011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M294.18,-35.5C294.69,-31.16 295.3,-26.09 295.82,-21.74\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"333\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"333\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">11</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M333,-70.74C333,-66.57 333,-61.76 333,-57.6\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"381\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"381\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M340.5,-75.75C349.46,-69.47 364.38,-59.03 373.39,-52.72\"/>\n",
       "</g>\n",
       "<!-- 1100 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>1100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">11</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1100 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>110&#45;&#45;1100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M331.82,-35.5C331.31,-31.16 330.7,-26.09 330.18,-21.74\"/>\n",
       "</g>\n",
       "<!-- 1101 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>1101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"361\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"361\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1101 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>110&#45;&#45;1101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M339.21,-37.94C343.85,-32.37 350.17,-24.8 354.8,-19.24\"/>\n",
       "</g>\n",
       "<!-- 1110 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>1110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"393\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"393\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1110 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>111&#45;&#45;1110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M384.41,-35.86C386.07,-31.21 388.06,-25.63 389.71,-21.01\"/>\n",
       "</g>\n",
       "<!-- 20 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>20</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 2&#45;&#45;20 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>2&#45;&#45;20</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-106.34C425,-102.17 425,-97.36 425,-93.2\"/>\n",
       "</g>\n",
       "<!-- 21 -->\n",
       "<g id=\"node32\" class=\"node\">\n",
       "<title>21</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 2&#45;&#45;21 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>2&#45;&#45;21</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M431.48,-109.8C437.03,-103.97 444.98,-95.62 450.53,-89.79\"/>\n",
       "</g>\n",
       "<!-- 200 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>200</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 20&#45;&#45;200 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>20&#45;&#45;200</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-70.74C425,-66.57 425,-61.76 425,-57.6\"/>\n",
       "</g>\n",
       "<!-- 2000 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>2000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">8</text>\n",
       "</g>\n",
       "<!-- 200&#45;&#45;2000 -->\n",
       "<g id=\"edge31\" class=\"edge\">\n",
       "<title>200&#45;&#45;2000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-35.14C425,-30.97 425,-26.16 425,-22\"/>\n",
       "</g>\n",
       "<!-- 210 -->\n",
       "<g id=\"node33\" class=\"node\">\n",
       "<title>210</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 21&#45;&#45;210 -->\n",
       "<g id=\"edge32\" class=\"edge\">\n",
       "<title>21&#45;&#45;210</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-70.74C457,-66.57 457,-61.76 457,-57.6\"/>\n",
       "</g>\n",
       "<!-- 2100 -->\n",
       "<g id=\"node34\" class=\"node\">\n",
       "<title>2100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 210&#45;&#45;2100 -->\n",
       "<g id=\"edge33\" class=\"edge\">\n",
       "<title>210&#45;&#45;2100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-35.14C457,-30.97 457,-26.16 457,-22\"/>\n",
       "</g>\n",
       "<!-- 2101 -->\n",
       "<g id=\"node35\" class=\"node\">\n",
       "<title>2101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"489\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"489\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 210&#45;&#45;2101 -->\n",
       "<g id=\"edge34\" class=\"edge\">\n",
       "<title>210&#45;&#45;2101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M463.48,-38.6C469.03,-32.77 476.98,-24.42 482.53,-18.59\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107e06210>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "root = build_my_tree(num_li)\n",
    "print(alpha_beta_pruning(root, False))\n",
    "root.draw_tree(False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6. Experiment with random trees"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(5, [1, 0, 0, 0])\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"506pt\" height=\"172pt\"\n",
       " viewBox=\"0.00 0.00 506.00 172.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 168)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-168 502,-168 502,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"317\" cy=\"-153.2\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"317\" y=\"-149.32\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M308.28,-149.67C286.78,-143.45 231.42,-127.41 209.82,-121.15\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"317\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"317\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M317,-141.94C317,-137.77 317,-132.96 317,-128.8\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>2</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;2 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>root&#45;&#45;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M325.84,-149.45C346.19,-143.12 396.16,-127.57 416.34,-121.29\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"121\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M192.44,-113C176.96,-106.5 144.52,-92.88 129.26,-86.47\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-106.34C201,-102.17 201,-97.36 201,-93.2\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M112.44,-77.4C96.96,-70.9 64.52,-57.28 49.26,-50.87\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"121\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"121\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M121,-70.74C121,-66.57 121,-61.76 121,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0000 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>0000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"9\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"9\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">10</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0000 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>000&#45;&#45;0000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.52,-38.6C28.97,-32.77 21.02,-24.42 15.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0001 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>0001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0001 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>000&#45;&#45;0001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M41,-35.14C41,-30.97 41,-26.16 41,-22\"/>\n",
       "</g>\n",
       "<!-- 0010 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>0010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"73\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"73\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0010 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>001&#45;&#45;0010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M113.5,-40.15C104.54,-33.87 89.62,-23.43 80.61,-17.12\"/>\n",
       "</g>\n",
       "<!-- 0011 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>0011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"105\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"105\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0011 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>001&#45;&#45;0011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M116.8,-36.57C114.43,-31.59 111.48,-25.4 109.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0012 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>0012</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"137\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"137\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0012 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>001&#45;&#45;0012</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M125.2,-36.57C127.57,-31.59 130.52,-25.4 132.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0013 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>0013</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0013 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>001&#45;&#45;0013</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M128.5,-40.15C137.46,-33.87 152.38,-23.43 161.39,-17.12\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-70.74C201,-66.57 201,-61.76 201,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0100 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>0100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0100 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>010&#45;&#45;0100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-35.14C201,-30.97 201,-26.16 201,-22\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"293\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"293\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M311.43,-108.81C307.56,-103.38 302.41,-96.17 298.54,-90.75\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"333\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"333\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M321.2,-107.77C323.57,-102.79 326.52,-96.6 328.88,-91.65\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"245\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"245\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M285.5,-75.75C276.54,-69.47 261.62,-59.03 252.61,-52.72\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"293\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"293\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M293,-70.74C293,-66.57 293,-61.76 293,-57.6\"/>\n",
       "</g>\n",
       "<!-- 1000 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>1000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"233\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"233\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1000 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>100&#45;&#45;1000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M241.59,-35.86C239.93,-31.21 237.94,-25.63 236.29,-21.01\"/>\n",
       "</g>\n",
       "<!-- 1010 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>1010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1010 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>101&#45;&#45;1010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M286.79,-37.94C282.15,-32.37 275.83,-24.8 271.2,-19.24\"/>\n",
       "</g>\n",
       "<!-- 1011 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>1011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1011 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>101&#45;&#45;1011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M294.18,-35.5C294.69,-31.16 295.3,-26.09 295.82,-21.74\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"333\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"333\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M333,-70.74C333,-66.57 333,-61.76 333,-57.6\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"381\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"381\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M340.5,-75.75C349.46,-69.47 364.38,-59.03 373.39,-52.72\"/>\n",
       "</g>\n",
       "<!-- 1100 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>1100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1100 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>110&#45;&#45;1100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M331.82,-35.5C331.31,-31.16 330.7,-26.09 330.18,-21.74\"/>\n",
       "</g>\n",
       "<!-- 1101 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>1101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"361\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"361\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1101 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>110&#45;&#45;1101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M339.21,-37.94C343.85,-32.37 350.17,-24.8 354.8,-19.24\"/>\n",
       "</g>\n",
       "<!-- 1110 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>1110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"393\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"393\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1110 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>111&#45;&#45;1110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M384.41,-35.86C386.07,-31.21 388.06,-25.63 389.71,-21.01\"/>\n",
       "</g>\n",
       "<!-- 20 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>20</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 2&#45;&#45;20 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>2&#45;&#45;20</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-106.34C425,-102.17 425,-97.36 425,-93.2\"/>\n",
       "</g>\n",
       "<!-- 21 -->\n",
       "<g id=\"node32\" class=\"node\">\n",
       "<title>21</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 2&#45;&#45;21 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>2&#45;&#45;21</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M431.48,-109.8C437.03,-103.97 444.98,-95.62 450.53,-89.79\"/>\n",
       "</g>\n",
       "<!-- 200 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>200</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 20&#45;&#45;200 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>20&#45;&#45;200</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-70.74C425,-66.57 425,-61.76 425,-57.6\"/>\n",
       "</g>\n",
       "<!-- 2000 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>2000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">4</text>\n",
       "</g>\n",
       "<!-- 200&#45;&#45;2000 -->\n",
       "<g id=\"edge31\" class=\"edge\">\n",
       "<title>200&#45;&#45;2000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M425,-35.14C425,-30.97 425,-26.16 425,-22\"/>\n",
       "</g>\n",
       "<!-- 210 -->\n",
       "<g id=\"node33\" class=\"node\">\n",
       "<title>210</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 21&#45;&#45;210 -->\n",
       "<g id=\"edge32\" class=\"edge\">\n",
       "<title>21&#45;&#45;210</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-70.74C457,-66.57 457,-61.76 457,-57.6\"/>\n",
       "</g>\n",
       "<!-- 2100 -->\n",
       "<g id=\"node34\" class=\"node\">\n",
       "<title>2100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 210&#45;&#45;2100 -->\n",
       "<g id=\"edge33\" class=\"edge\">\n",
       "<title>210&#45;&#45;2100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-35.14C457,-30.97 457,-26.16 457,-22\"/>\n",
       "</g>\n",
       "<!-- 2101 -->\n",
       "<g id=\"node35\" class=\"node\">\n",
       "<title>2101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"489\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"489\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 210&#45;&#45;2101 -->\n",
       "<g id=\"edge34\" class=\"edge\">\n",
       "<title>210&#45;&#45;2101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M463.48,-38.6C469.03,-32.77 476.98,-24.42 482.53,-18.59\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107e07990>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# You can repeat this random experiment as many times as you want.\n",
    "\n",
    "root = build_my_tree()\n",
    "print(alpha_beta_pruning(root, True))\n",
    "root.draw_tree(True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(5, [1, 1, 0, 0])\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 8.1.0 (20230707.0739)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"506pt\" height=\"172pt\"\n",
       " viewBox=\"0.00 0.00 506.00 172.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 168)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-168 502,-168 502,4 -4,4\"/>\n",
       "<!-- root -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>root</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-153.2\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-149.32\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 0 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>0</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"185\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"185\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;0 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>root&#45;&#45;0</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M256.44,-148.6C240.96,-142.1 208.52,-128.48 193.26,-122.07\"/>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>1</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"313\" cy=\"-117.6\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"313\" y=\"-113.72\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- root&#45;&#45;1 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>root&#45;&#45;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M272.5,-146.95C281.46,-140.67 296.38,-130.23 305.39,-123.92\"/>\n",
       "</g>\n",
       "<!-- 00 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>00</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"89\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"89\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;00 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>0&#45;&#45;00</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M176.48,-113.62C158.31,-107.26 115.92,-92.42 97.64,-86.02\"/>\n",
       "</g>\n",
       "<!-- 01 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>01</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"185\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"185\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 0&#45;&#45;01 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>0&#45;&#45;01</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M185,-106.34C185,-102.17 185,-97.36 185,-93.2\"/>\n",
       "</g>\n",
       "<!-- 000 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;000 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>00&#45;&#45;000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M81.5,-75.75C72.54,-69.47 57.62,-59.03 48.61,-52.72\"/>\n",
       "</g>\n",
       "<!-- 001 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"89\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"89\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 00&#45;&#45;001 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>00&#45;&#45;001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M89,-70.74C89,-66.57 89,-61.76 89,-57.6\"/>\n",
       "</g>\n",
       "<!-- 0000 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>0000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"9\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"9\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">9</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0000 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>000&#45;&#45;0000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.52,-38.6C28.97,-32.77 21.02,-24.42 15.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0001 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>0001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"41\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"41\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 000&#45;&#45;0001 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>000&#45;&#45;0001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M41,-35.14C41,-30.97 41,-26.16 41,-22\"/>\n",
       "</g>\n",
       "<!-- 0010 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>0010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"73\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"73\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">6</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0010 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>001&#45;&#45;0010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M84.8,-36.57C82.43,-31.59 79.48,-25.4 77.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 0011 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>0011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"105\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"105\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 001&#45;&#45;0011 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>001&#45;&#45;0011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M93.2,-36.57C95.57,-31.59 98.52,-25.4 100.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 010 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;010 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>01&#45;&#45;010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M180.8,-72.17C178.43,-67.19 175.48,-61 173.12,-56.05\"/>\n",
       "</g>\n",
       "<!-- 011 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">max</text>\n",
       "</g>\n",
       "<!-- 01&#45;&#45;011 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>01&#45;&#45;011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M189.2,-72.17C191.57,-67.19 194.52,-61 196.88,-56.05\"/>\n",
       "</g>\n",
       "<!-- 0100 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>0100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"137\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"137\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">2</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0100 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>010&#45;&#45;0100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M162.52,-38.6C156.97,-32.77 149.02,-24.42 143.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 0101 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>0101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"169\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"169\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">7</text>\n",
       "</g>\n",
       "<!-- 010&#45;&#45;0101 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>010&#45;&#45;0101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M169,-35.14C169,-30.97 169,-26.16 169,-22\"/>\n",
       "</g>\n",
       "<!-- 0110 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>0110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"201\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"201\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 011&#45;&#45;0110 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>011&#45;&#45;0110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M201,-35.14C201,-30.97 201,-26.16 201,-22\"/>\n",
       "</g>\n",
       "<!-- 0111 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>0111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"233\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"233\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 011&#45;&#45;0111 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>011&#45;&#45;0111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M207.48,-38.6C213.03,-32.77 220.98,-24.42 226.53,-18.59\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>10</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"313\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"313\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;10 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>1&#45;&#45;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M313,-106.34C313,-102.17 313,-97.36 313,-93.2\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>11</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"409\" cy=\"-82\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"409\" y=\"-78.12\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 1&#45;&#45;11 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>1&#45;&#45;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M321.52,-113.62C339.69,-107.26 382.08,-92.42 400.36,-86.02\"/>\n",
       "</g>\n",
       "<!-- 100 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;100 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>10&#45;&#45;100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M308.8,-72.17C306.43,-67.19 303.48,-61 301.12,-56.05\"/>\n",
       "</g>\n",
       "<!-- 101 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 10&#45;&#45;101 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>10&#45;&#45;101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M317.2,-72.17C319.57,-67.19 322.52,-61 324.88,-56.05\"/>\n",
       "</g>\n",
       "<!-- 1000 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>1000</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"265\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"265\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1000 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>100&#45;&#45;1000</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M290.52,-38.6C284.97,-32.77 277.02,-24.42 271.47,-18.59\"/>\n",
       "</g>\n",
       "<!-- 1001 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>1001</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"297\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"297\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 100&#45;&#45;1001 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>100&#45;&#45;1001</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M297,-35.14C297,-30.97 297,-26.16 297,-22\"/>\n",
       "</g>\n",
       "<!-- 1010 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>1010</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"329\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"329\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">3</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1010 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>101&#45;&#45;1010</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M329,-35.14C329,-30.97 329,-26.16 329,-22\"/>\n",
       "</g>\n",
       "<!-- 1011 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>1011</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"361\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"361\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 101&#45;&#45;1011 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>101&#45;&#45;1011</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M335.48,-38.6C341.03,-32.77 348.98,-24.42 354.53,-18.59\"/>\n",
       "</g>\n",
       "<!-- 110 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"409\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"409\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;110 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>11&#45;&#45;110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M409,-70.74C409,-66.57 409,-61.76 409,-57.6\"/>\n",
       "</g>\n",
       "<!-- 111 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-46.4\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-42.52\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 11&#45;&#45;111 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>11&#45;&#45;111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M416.5,-75.75C425.46,-69.47 440.38,-59.03 449.39,-52.72\"/>\n",
       "</g>\n",
       "<!-- 1100 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>1100</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"393\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"393\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1100 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>110&#45;&#45;1100</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M404.8,-36.57C402.43,-31.59 399.48,-25.4 397.12,-20.45\"/>\n",
       "</g>\n",
       "<!-- 1101 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>1101</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"425\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"425\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">1</text>\n",
       "</g>\n",
       "<!-- 110&#45;&#45;1101 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>110&#45;&#45;1101</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M413.2,-36.57C415.57,-31.59 418.52,-25.4 420.88,-20.45\"/>\n",
       "</g>\n",
       "<!-- 1110 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>1110</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"457\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"457\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\">5</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1110 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>111&#45;&#45;1110</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457,-35.14C457,-30.97 457,-26.16 457,-22\"/>\n",
       "</g>\n",
       "<!-- 1111 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>1111</title>\n",
       "<ellipse fill=\"none\" stroke=\"black\" cx=\"489\" cy=\"-10.8\" rx=\"9\" ry=\"10.8\"/>\n",
       "<text text-anchor=\"middle\" x=\"489\" y=\"-6.92\" font-family=\"Times,serif\" font-size=\"10.00\" fill=\"red\">min</text>\n",
       "</g>\n",
       "<!-- 111&#45;&#45;1111 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>111&#45;&#45;1111</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M463.48,-38.6C469.03,-32.77 476.98,-24.42 482.53,-18.59\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.sources.Source at 0x107e0cc90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# You can repeat this random experiment as many times as you want.\n",
    "\n",
    "root = build_bin_tree(4)\n",
    "print(alpha_beta_pruning(root, False))\n",
    "root.draw_tree(False)\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "mlenv",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}