diff --git a/loopy/tools.py b/loopy/tools.py
index 8e8453602..2a119eae2 100644
--- a/loopy/tools.py
+++ b/loopy/tools.py
@@ -1004,7 +1004,7 @@ def from_root(root: T):
         return Tree(pmap({root: frozenset()}),
                     pmap({root: None}))
 
-    @property
+    @cached_property
     def root(self) -> T:
         guess = set(self._child_to_parent).pop()
         while self.parent(guess) is not None:
@@ -1012,6 +1012,7 @@ def root(self) -> T:
 
         return guess
 
+    @memoize_method
     def ancestors(self, node: T) -> "FrozenSet[T]":
         """
         Returns a :class:`frozenset` of nodes that are ancestors of *node*.
@@ -1039,6 +1040,7 @@ def children(self, node: T) -> "FrozenSet[T]":
 
         return self._parent_to_children[node]
 
+    @memoize_method
     def depth(self, node: T) -> int:
         if not self.is_a_node(node):
             raise ValueError(f"'{node}' not in tree.")
diff --git a/loopy/transform/loop_fusion.py b/loopy/transform/loop_fusion.py
index 16df080f7..584fc3f50 100644
--- a/loopy/transform/loop_fusion.py
+++ b/loopy/transform/loop_fusion.py
@@ -275,18 +275,17 @@ def _get_ldg_nodes_from_loopy_insn(kernel, insn, candidates, non_candidates,
                 for candidate in ((insn.within_inames
                                    | insn.reduction_inames())
                                   & candidates)]
-    elif {loop_nest
-          for loop_nest in non_candidates
-          if (loop_nest & insn.within_inames)}:
+    else:
         non_candidate, = {loop_nest
                           for loop_nest in non_candidates
                           if (loop_nest & insn.within_inames)}
 
-        return [NonCandidateLoop(non_candidate)]
-    else:
-        assert ((insn.within_inames & just_outer_loop_nest)
-                or (insn.within_inames == just_outer_loop_nest))
-        return [OuterLoopNestStatement(insn.id)]
+        if non_candidate:
+            return [NonCandidateLoop(non_candidate)]
+        else:
+            assert ((insn.within_inames & just_outer_loop_nest)
+                    or (insn.within_inames == just_outer_loop_nest))
+            return [OuterLoopNestStatement(insn.id)]
 
 
 @memoize_on_first_arg