Solution Explanation

This problem asks to find the Lowest Common Ancestor (LCA) of multiple nodes in a binary tree. The standard LCA algorithms for two nodes need modification to handle multiple nodes. The efficient solution uses Depth-First Search (DFS) and a set to track the nodes we're looking for.

Approach:

1. Create a Set: We first create a set s containing the values of all nodes in the nodes array. This allows for O(1) lookup time to check if a node is one of the target nodes.

2. Depth-First Search (DFS): A recursive DFS function dfs is implemented. This function traverses the tree.

3. Base Cases:

   • If the current node (root) is null (we've reached the end of a branch), it returns null.
   • If the current node's value is present in the set s (it's one of the target nodes), it returns the current node itself.

4. Recursive Calls: The function recursively calls itself on the left and right subtrees.

5. LCA Logic:

   • If both recursive calls (left and right) return non-null values, it means we've found at least one target node in the left subtree and at least one in the right subtree. Therefore, the current node (root) is their LCA, and it's returned.
   • If only one of the recursive calls returns a non-null value, it means all target nodes are found in either the left or right subtree, and that subtree's result is returned.
   • If both return null, it means no target nodes were found in the subtree. null is returned in this case.

Time Complexity Analysis:

The time complexity is O(N), where N is the number of nodes in the binary tree. In the worst case, the DFS function visits each node in the tree once. The set creation takes O(K) time where K is the number of nodes in the nodes array, which is much smaller than N in most cases. Therefore, O(N) dominates the complexity.

Space Complexity Analysis:

The space complexity is O(H + K) in the worst case, where H is the height of the tree (for the recursion stack in DFS) and K is the number of nodes in the nodes array (for the set s). In a balanced tree, H is O(log N), and in a skewed tree, H is O(N). Again, K is generally much smaller than N.

Code in Different Languages:

The code provided in the original response is already well-structured and efficient. The comments within the code explain the logic adequately. No further improvements are needed. The code is given for Python, Java, C++, and JavaScript. All examples follow the described approach. The key improvements are already present: using a set for efficient node lookup and a concise recursive DFS.