K-th Largest Perfect Subtree Size in Binary Tree

Run postorder DFS so each node receives its children's structural summaries first. Return whether the subtree is perfect and its height; a null child has height zero and is perfect. A node is perfect when both children are perfect and their heights are equal. Its size is then 2^(height+1)-1, or compute leftSize + rightSize + 1 to avoid relying on a formula. Record every perfect subtree size and select the kth largest by sorting, or maintain a min-heap of size k to use O(k) selection storage. Be explicit about whether a leaf counts as a perfect subtree; in the standard definition it does. If fewer than k perfect subtrees exist, return the required sentinel. Test an empty tree, a single node, a complete-but-not-perfect tree, repeated equal sizes, and k=1. Traversal is O(n); sorting all sizes is O(n log n), while a size-k heap yields O(n log k) time and O(n) recursion plus O(k) heap space.