110. Balanced Binary Tree

题目：

Given a binary tree, determine if it is height-balanced.

For this problem, a height-balanced binary tree is defined as a binary tree in which the depth of the two subtrees of every node never differ by more than 1.

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 Tree Depth-first Search 

链接： http://leetcode.com/problems/balanced-binary-tree/

题解：

求一棵树是否是平衡树。根据定义计算两个子节点的深度，当深度差大于1时返回false，依然是DFS。

Time Complexity - O(n)， Space Complexity - O(n)

public class Solution {
    public boolean isBalanced(TreeNode root) {
        return getDepth(root) != -1;
    }
    
    private int getDepth(TreeNode node){
        if(node == null)
            return 0;
        int leftDepth = getDepth(node.left);
        int rightDepth = getDepth(node.right);
        if(leftDepth  == -1 || rightDepth == -1 || Math.abs(leftDepth - rightDepth) > 1)
            return -1;
        return 1 + Math.max(leftDepth, rightDepth);
    }
}

Update:

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     int val;
 *     TreeNode left;
 *     TreeNode right;
 *     TreeNode(int x) { val = x; }
 * }
 */
public class Solution {
    public boolean isBalanced(TreeNode root) {
        if(root == null)
            return true;
        int left = getLength(root.left);
        int right = getLength(root.right);    
        if(Math.abs(left - right) > 1)
            return false;
        return isBalanced(root.left) && isBalanced(root.right);
    }
    
    private int getLength(TreeNode root) {
        if(root == null)
            return 0;
        return 1 + Math.max(getLength(root.left), getLength(root.right));
    }
}

二刷:

仔细看了一下discuss，有top-down和bottom-up两种，top-down的话要多search到不少东西。 bottom-up solution才是O(n)

Java:

Top-down，就是从上到下，先求出节点的depth，然后再看子节点是否balance。这里其实求depth已经计算过节点了，然后还要做两个isBalanced，多做了运算。

Time Complexity - O(n²)， Space Complexity - O(n)

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     int val;
 *     TreeNode left;
 *     TreeNode right;
 *     TreeNode(int x) { val = x; }
 * }
 */
public class Solution {
    public boolean isBalanced(TreeNode root) {
        if (root == null) {
            return true;
        }
        return (Math.abs(getDepth(root.left) - getDepth(root.right)) < 2) && isBalanced(root.left) && isBalanced(root.right);
    }
    
    private int getDepth(TreeNode root) {
        if (root == null) {
            return 0;
        }
        return 1 + Math.max(getDepth(root.left), getDepth(root.right));
    }
}

bottom-up: 在不满足条件的时候就直接剪枝，提高了运算效率。

Time Complexity - O(n)， Space Complexity - O(n)

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     int val;
 *     TreeNode left;
 *     TreeNode right;
 *     TreeNode(int x) { val = x; }
 * }
 */
public class Solution {
    public boolean isBalanced(TreeNode root) {
       return getDepth(root) != -1;
    }
    
    private int getDepth(TreeNode root) {
        if (root == null) {
            return 0;
        }
        int left = getDepth(root.left);
        if (left == -1) {
            return -1;
        }
        int right = getDepth(root.right);
        if (right == -1) {
            return -1;
        }
        if (Math.abs(left - right) > 1) {
            return -1;
        }
        return 1 + Math.max(left, right);
    }
}

三刷：

这道题在Microsot的online test里遇见了原题。

我们可以使用自底向上的方法，用-1作为false的indicator，将不满足条件的dfs直接剪掉。最后Time Complexity就是O(n)， Space Complexity也是O(n)。

Java:

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     int val;
 *     TreeNode left;
 *     TreeNode right;
 *     TreeNode(int x) { val = x; }
 * }
 */
public class Solution {
    public boolean isBalanced(TreeNode root) {
        return getDepth(root) != -1;
    }
    
    private int getDepth(TreeNode root) {
        if (root == null) {
            return 0;
        } 
        int leftDepth = getDepth(root.left);
        if (leftDepth == -1) {
            return -1;
        }
        int rightDepth = getDepth(root.right);
        if (rightDepth == -1) {
            return -1;    
        }
        if (Math.abs(leftDepth - rightDepth) > 1) {
            return -1;
        }
        return 1 + Math.max(leftDepth, rightDepth);
    }
}

Updated:

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     int val;
 *     TreeNode left;
 *     TreeNode right;
 *     TreeNode(int x) { val = x; }
 * }
 */
public class Solution {
    public boolean isBalanced(TreeNode root) {
        return getDepth(root) != -1;
    }
    
    private int getDepth(TreeNode root) {
        if (root == null) return 0;
        int left = getDepth(root.left);
        if (left == -1) return -1;
        int right = getDepth(root.right);
        if (right == -1) return -1;
        if (Math.abs(left - right) > 1) return -1;
        return 1 + Math.max(left, right);
    }
}

题外话:

这种扩展方法，就像你做网站里有一个field是checkbox yes / no， 要把它扩展为一个dropdownlist   "entry#1, entry#2, entry#3"一样。 做有些算法题跟中学做几何题一样，关键是如何做辅助线，辅助函数，做得好自然解得快。 

Reference:

https://leetcode.com/discuss/22898/the-bottom-up-o-n-solution-would-be-better

https://leetcode.com/discuss/3931/can-we-have-a-better-solution

https://leetcode.com/discuss/29893/solution-height-every-recursion-avoid-further-useless-search

https://leetcode.com/discuss/12331/accepted-o-n-solution

https://leetcode.com/discuss/28162/java-o-n-solution-based-on-maximum-depth-of-binary-tree