Smallest Subtree with all the Deepest Nodes

Problem Description

Given the root of a binary tree, the depth of each node is the shortest distance to the root. Return the smallest subtree such that it contains all the deepest nodes in the original tree. A node is called the deepest if it has the largest depth possible among any node in the entire tree. The subtree of a node is a tree consisting of that node, plus the set of all descendants of that node.

Key Insights

• A node's depth is defined as the shortest distance from the root node.
• The problem requires identifying the deepest nodes in the tree.
• The smallest subtree containing all deepest nodes will be the Lowest Common Ancestor (LCA) of those nodes.
• We can perform a depth-first search (DFS) or breadth-first search (BFS) to find the deepest nodes and their ancestor.

Space and Time Complexity

Time Complexity: O(N) - where N is the number of nodes in the tree, since we may need to visit each node once. Space Complexity: O(H) - where H is the height of the tree, which is the space used by the call stack in the case of DFS.

Solution

To solve this problem, we can use a depth-first search (DFS) approach that allows us to traverse the tree to find the deepest nodes and their common ancestor. We will maintain two variables: one to track the maximum depth encountered so far and another to keep track of the node that becomes the smallest subtree that includes all deepest nodes. The algorithm follows these steps:

1. Traverse the tree using DFS.
2. For each node, calculate its depth.
3. If the current node's depth is greater than the maximum depth found so far, update the maximum depth and set the current node as the LCA.
4. If the current node's depth equals the maximum depth, continue traversing upwards to find the LCA.
5. Return the LCA as the smallest subtree containing all deepest nodes.

Code Solutions