Count Complete Tree Nodes

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

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.

Analysis

Note:

1. 1<<n == 2^n
2. 一个complete tree的nodes数为2^height-1
3. 假设root的height为h，那左边height数一定是h-1
4. 如果左右height数相同为h-1
   1. 左边一定是complete tree
   2. nodes数为以下相加
      1. root:1
      2. left sub tree: 2^(h-1) - 1
      3. right sub tree: nodes(right sub tree)
5. 如果左右height数不同，右边height数一定是h-2
   1. 右边一定是height为h-2的complete tree
   2. nodes数为以下相加
      1. root: 1
      2. left sub tree: nodes(left sub tree)
      3. right sub tree: 2^(h-2) -1

Code

public class Solution {
    public int countNodes(TreeNode root) {
        int h = height(root);
        if (h==0) {return 0;}
        //如果左右subtree height想同，说明左subtree一定是complete tree,那么只要把
        return height(root.right) == h-1 ? (1<<h-1) + countNodes(root.right) : (1<<h-2) + countNodes(root.left);
    }

    //helper funciton to count the height of a tree
    private int height(TreeNode root){
        return root==null? 0 : 1+height(root.left);
    }
}

Reference

Leetcode