951. Flip Equivalent Binary Trees

Difficulty: Medium

Topics: Tree, Depth-First Search, Binary Tree

For a binary tree T, we can define a flip operation as follows: choose any node, and swap the left and right child subtrees.

A binary tree X is flip equivalent to a binary tree Y if and only if we can make X equal to Y after some number of flip operations.

Given the roots of two binary trees root1 and root2, return true if the two trees are flip equivalent or false otherwise.

Example 1:

Input: root1 = [1,2,3,4,5,6,null,null,null,7,8], root2 = [1,3,2,null,6,4,5,null,null,null,null,8,7]
Output: true
Explanation: We flipped at nodes with values 1, 3, and 5.

Example 2:

Input: root1 = [], root2 = []
Output: true

Example 3:

Input: root1 = [], root2 = [1]
Output: false

Constraints:

The number of nodes in each tree is in the range [0, 100].
Each tree will have unique node values in the range [0, 99].

Solution:

We can use a recursive depth-first search (DFS). The idea is that two trees are flip equivalent if their root values are the same, and the subtrees are either the same (without any flips) or they become the same after flipping the left and right children at some nodes.

Plan:

Base Cases:
- If both root1 and root2 are null, they are trivially flip equivalent.
- If only one of them is null, they can't be equivalent.
- If the root values of root1 and root2 are different, they can't be equivalent.
Recursive Case:
- Recursively check two possibilities:
  1. The left subtree of root1 is flip equivalent to the left subtree of root2, and the right subtree of root1 is flip equivalent to the right subtree of root2 (i.e., no flip).
  2. The left subtree of root1 is flip equivalent to the right subtree of root2, and the right subtree of root1 is flip equivalent to the left subtree of root2 (i.e., flip the children).

Let's implement this solution in PHP: 951. Flip Equivalent Binary Trees

<?php
// Definition for a binary tree node.
class TreeNode {
    public $val;
    public $left;
    public $right;
    function __construct($val = 0, $left = null, $right = null) {
        $this->val = $val;
        $this->left = $left;
        $this->right = $right;
    }
}

/**
 * @param TreeNode $root1
 * @param TreeNode $root2
 * @return Boolean
 */
function flipEquiv($root1, $root2) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Example usage:
$root1 = new TreeNode(1,
    new TreeNode(2, new TreeNode(4), new TreeNode(5, new TreeNode(7), new TreeNode(8))),
    new TreeNode(3, new TreeNode(6), null)
);

$root2 = new TreeNode(1,
    new TreeNode(3, null, new TreeNode(6)),
    new TreeNode(2, new TreeNode(4), new TreeNode(5, new TreeNode(8), new TreeNode(7)))
);

var_dump(flipEquiv($root1, $root2)); // Output: bool(true)
?>

Explanation:

TreeNode Class: The TreeNode class represents a node in the binary tree, with a constructor to initialize the node's value, left child, and right child.
flipEquiv Function:
- The base cases handle when both nodes are null, when one is null, or when the values do not match.
- The recursive case checks both possibilities (no flip vs. flip), ensuring the subtrees are flip equivalent under either condition.

Time Complexity:

The function checks every node in both trees, and each recursive call processes two subtrees. Therefore, the time complexity is O(N), where N is the number of nodes in the tree.

Space Complexity:

The space complexity is O(H), where H is the height of the tree, because of the recursive stack. In the worst case (for a skewed tree), this could be O(N).

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