LCA in Binary Tree

Problem Overview

The problem presented by LeetCode is to find the "Lowest Common Ancestor of a Binary Tree." In this task, you are to determine the lowest common ancestor (LCA) of two given nodes in a binary tree.

Understanding the Problem

In a binary tree, the lowest common ancestor between two nodes, p and q, is defined as the lowest node in the tree that has both p and q as descendants (where we allow a node to be a descendant of itself). The binary tree is not a binary search tree, which means that the values of the nodes do not follow the properties of a BST.

To provide clarity, consider the following example:

        3
       / \
      5   1
     / \ / \
    6  2 0  8
      / \
     7   4

Given the nodes p = 5 and q = 1, the LCA is 3. For p = 5 and q = 4, the LCA is 5, as 5 is a descendant of itself.

Key DSA Concepts and Theory

Binary Tree

A binary tree is a hierarchical structure where each node has at most two child nodes referred to as the left child and right child.

Lowest Common Ancestor (LCA)

Finding the Lowest Common Ancestor in a binary tree requires navigating the tree to find the shared ancestor node that is nearest to the leaf nodes. This typically involves recursive traversal, a fundamental concept related to trees.

Recursion in Trees

Recursion is a method of solving a problem where the solution depends on solutions to smaller instances of the same problem. It's particularly useful with tree structures, as it allows us to traverse nodes efficiently.

Solution Approach

The solution follows a recursive approach that employs depth-first search (DFS). The idea is to recursively visit nodes starting from the root. The algorithm utilizes three main conditions:

1. If the current node is None, return None.
2. If the current node matches either p or q, return the current node.
3. Recursively search the left and right subtree for the nodes p and q.

Steps:

1. Start with the root of the tree.
2. Recursively check the left subtree.
3. Recursively check the right subtree.
4. If both subtrees contain one of the targets (p or q), then the current node is their LCA.
5. Otherwise, return the non-null value from either the left or right subtree as it indicates the path to either p or q.

C++ Code:

class Solution {
public:
    TreeNode* lowestCommonAncestor(TreeNode* root, TreeNode* p, TreeNode* q) {
        if (root == nullptr || root == p || root == q) {
            return root;
        }
        
        TreeNode* left = lowestCommonAncestor(root->left, p, q);
        TreeNode* right = lowestCommonAncestor(root->right, p, q);
        
        if (left != nullptr && right != nullptr) {
            return root;
        }
        return left != nullptr ? left : right;
    }
};

Java Code:

public class Solution {
    public TreeNode lowestCommonAncestor(TreeNode root, TreeNode p, TreeNode q) {
        if (root == null || root == p || root == q) {
            return root;
        }

        TreeNode left = lowestCommonAncestor(root.left, p, q);
        TreeNode right = lowestCommonAncestor(root.right, p, q);

        if (left != null && right != null) {
            return root;
        }
        return left != null ? left : right;
    }
}

Time and Space Complexity Analysis

• Time Complexity: O(N), where N is the number of nodes in the binary tree. This complexity arises because, in the worst-case scenario, every node in the tree needs to be visited.

• Space Complexity: O(H), where H is the height of the binary tree. This is due to the recursive stack that may be as deep as the height of the tree.

Common Mistakes to Avoid

• Confusing binary tree properties with binary search tree properties. The problem does not involve sorted properties.
• Forgetting to consider that either p or q can be the ancestor of the other. It’s essential to check if the current node directly corresponds to p or q.

Similar Problems on LeetCode

1. Lowest Common Ancestor of a Binary Search Tree - LeetCode 235
2. Smallest Subtree with all the Deepest Nodes - LeetCode 865

Additional Resources and References

• Introduction to Binary Trees from various online courses (Coursera, Udacity, etc.)
• Articles and videos on recursion and deeper exploration on how it affects tree traversal.
• Recommended book: "Cracking the Coding Interview" for detailed problems on trees and recursion.

This structured approach and understanding ensure a comprehensive solution to the problem of finding the Lowest Common Ancestor in a Binary Tree. Happy coding!