Solution Explanation

This problem asks to find the Lowest Common Ancestor (LCA) of two nodes, p and q, in a binary tree. The key challenge is handling the case where one or both of p and q might not exist in the tree.

The solution uses a Depth-First Search (DFS) approach. The core idea is to recursively traverse the tree. For each node, we check if it's p or q or if its subtrees contain p and q.

Algorithm:

1. Base Case: If the current node (root) is null, it means we've reached the end of a branch, so we return false (indicating that p and q weren't found in that branch).

2. Recursive Calls: We recursively call the dfs function on the left and right subtrees. l and r store the results of these recursive calls (true if p or q are found in the respective subtree, false otherwise).

3. LCA Check:

   • If both l and r are true, it means p and q are found in both the left and right subtrees. Therefore, the current node (root) is their LCA, and we update ans to root.
   • If either l or r is true, and the current node's value is equal to p.val or q.val, then the current node is either p or q (or both if p or q is an ancestor of the other), and it's also the LCA. We update ans accordingly.

4. Return Value: The function returns true if p or q (or both) are found in the subtree rooted at the current node; otherwise, it returns false.

5. Result: The global variable ans stores the LCA after the entire tree has been traversed.

Time Complexity Analysis:

The DFS function visits each node in the tree at most once. Therefore, the time complexity is O(N), where N is the number of nodes in the binary tree.

Space Complexity Analysis:

The space complexity is determined by the recursion depth, which in the worst case (a skewed tree) can be O(N) to store the call stack. In a balanced tree, the space complexity would be O(log N).

Code Explanation (Python):

The Python code directly implements the algorithm described above. The dfs function performs the recursive traversal, and the main function simply initializes the ans variable and calls dfs with the root node. The ans variable is declared as nonlocal so that it can be accessed and modified within the dfs function.

Code Explanation (Java, C++, JavaScript):

The Java, C++, and JavaScript code follow the same algorithm. They use a similar recursive approach and a global variable (ans) to store the LCA. Note that in Java and C++, we use a class member variable instead of nonlocal since there's no equivalent in those languages. The implementation structure is almost identical in terms of logic to the python code.