How to Calculate the Tilt of a Binary Tree - PHP Implementation Explained

Overview of Binary Trees

A binary tree is a commonly used data structure consisting of nodes, where each node can have at most two child nodes. Typically, the value of the left child is smaller than the value of the right child. Binary trees have wide applications in computer science, such as search binary trees, balanced binary trees, and heaps. This article explains how to calculate the tilt of a binary tree using PHP.

What is the Tilt of a Binary Tree?

The tilt of a binary tree refers to the absolute difference between the sum of values in its left and right subtrees for each node. The overall tilt of the binary tree is the sum of the tilts of all its nodes.

PHP Code Implementation

/**
 * Definition for a binary tree node
 * class TreeNode {
 *     public $val = null;
 *     public $left = null;
 *     public $right = null;
 *     function __construct($value) { $this->val = $value; }
 * }
 */
class Solution {
    public $res = 0;

    function findTilt($root) {
        $this->dfs($root);
        return $this->res;
    }

    // DFS to calculate the tilt of the binary tree
    function dfs($root) {
        if ($root == null) return 0;
        $left = $this->dfs($root->left);
        $right = $this->dfs($root->right);
        $this->res += abs($left - $right);
        return $left + $right + $root->val;
    }
}

The PHP code above defines a binary tree class and implements DFS (Depth-First Search) to traverse the tree and calculate the tilt of each node. The findTilt function returns the overall tilt of the tree, while the dfs function recursively traverses each node and returns the sum of its left and right subtrees.

Algorithm Analysis

To calculate the tilt of the binary tree, we need to traverse all the nodes and compute the absolute difference between the sum of values in their left and right subtrees. As we traverse, we accumulate the tilt for each node, and by the end, we have the total tilt of the entire binary tree.

We use a recursive approach to traverse the tree from bottom to top. For each node, we first compute the tilt of its left and right subtrees, and then add the absolute difference to the total tilt.

The time complexity of this algorithm is O(n), where n is the number of nodes in the binary tree.

Conclusion

This article explains how to calculate the tilt of a binary tree using PHP. By using a depth-first search algorithm, we recursively traverse each node, calculate the tilt, and accumulate it to get the total tilt of the tree. The time complexity of this algorithm is O(n), making it an efficient solution for the tilt problem in binary trees.