951. Flip Equivalent Binary Trees - Detailed Explanation

Problem Statement

You are given the roots of two binary trees. Two binary trees are considered flip equivalent if they are equal after performing any number of flip operations. A flip operation consists of swapping the left and right child of a node. Your task is to determine whether the two trees are flip equivalent.

For example, consider these two trees:

    Tree 1:          Tree 2:
        1                1
       / \              / \
      2   3            3   2
     /                    \
    4                      4

Even though the trees are structured differently, by flipping the children of the root node in one of the trees, they become identical. Therefore, these trees are flip equivalent.

Hints for the Approach

1. Recursive Comparison:
   Think about a recursive function that checks if two nodes are flip equivalent. At each node, consider both the scenario where you do not flip the children and the scenario where you do flip them.

2. Base Cases:

   • If both nodes are null, they are trivially equivalent.
   • If one is null and the other is not, or if the values at the nodes differ, the trees cannot be equivalent.

3. Two Possibilities:
   For non-null nodes with the same value, you need to check:

   • No Flip: The left child of the first tree is equivalent to the left child of the second tree, and the right child is equivalent to the right child.
   • Flip: The left child of the first tree is equivalent to the right child of the second tree, and the right child is equivalent to the left child.

Detailed Explanation of the Recursive Approach

Step 1: Base Cases

• Both Null: If both nodes are null, return true because two empty trees are equivalent.
• One Null: If one node is null and the other is not, return false.
• Value Mismatch: If the values of the nodes differ, return false.

Step 2: Recursive Check Without Flip

• Recursively check if the left child of the first node is flip equivalent to the left child of the second node and the right child of the first node is flip equivalent to the right child of the second node.

Step 3: Recursive Check With Flip

• Recursively check if the left child of the first node is flip equivalent to the right child of the second node and the right child of the first node is flip equivalent to the left child of the second node.

Step 4: Combine the Checks

• Return true if either the no-flip case or the flip case is true.

Complexity Analysis

• Time Complexity:
  In the worst case, every node is visited once and two recursive calls are made at each node. 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 due to the recursion stack. In the worst-case scenario (skewed tree), h can be O(n).