Count Complete Tree Nodes

Given a complete binary tree, count the number of nodes.

Note:

Definition of a complete binary tree from Wikipedia: In a complete binary tree every level, except possibly the last, is completely filled, and all nodes in the last level are as far left as possible. It can have between 1 and 2h nodes inclusive at the last level h.

Example:

Input: 
    1
   / \
  2   3
 / \  /
4  5 6

Output: 6

Explanation

The height of a tree can be found by just going left. Let a single node tree have height 0. Find the height h of the whole tree. If the whole tree is empty, i.e., has height -1, there are 0 nodes.

Otherwise check whether the height of the right subtree is just one less than that of the whole tree, meaning left and right subtree have the same height.

• If yes, then the last node on the last tree row is in the right subtree and the left subtree is a full tree of height h-1. So we take the 2^h-1 nodes of the left subtree plus the 1 root node plus recursively the number of nodes in the right subtree.

• If no, then the last node on the last tree row is in the left subtree and the right subtree is a full tree of height h-2. So we take the 2^(h-1)-1 nodes of the right subtree plus the 1 root node plus recursively the number of nodes in the left subtree.

Since I halve the tree in every recursive step, I have O(log(n)) steps. Finding a height costs O(log(n)). So overall O(log(n)^2).

class Solution {
    public int countNodes(TreeNode root) {
        if (root == null)
            return 0;

        int totalDepth = getDepth(root);
        int rightDepth = getDepth(root.right);
        // This means left subtree is full
        if (rightDepth + 1 == totalDepth)
            // 1st part is root and left subtree
            return (1 << (totalDepth - 1)) + countNodes(root.right);
        // This means right subtree is full
        // 2nd part is root and right subtree
        return (1 << (totalDepth - 2)) + countNodes(root.left);
    }

    private int getDepth(TreeNode root) {
        if (root == null)
            return 0;
        return 1 + getDepth(root.left);
    }
}