Smallest Subtree with all the Deepest Nodes
Given the root of a binary tree, the depth of each node is the shortest distance to the root. Find the node that is the lowest common ancestor of all the deepest nodes in the tree.
Why Interviewers Ask This
Meta interviewers use this problem to assess a candidate's mastery of tree recursion and depth-first search. It specifically tests the ability to handle post-order traversal logic where a node's value depends on its children's return values. The question evaluates if you can efficiently compute depths and identify the Lowest Common Ancestor without redundant traversals, demonstrating strong algorithmic optimization skills.
How to Answer This Question
1. Clarify requirements: Confirm if the tree is balanced or skewed and define 'deepest' as nodes with maximum depth from the root. 2. Propose a recursive strategy: Explain that a single DFS pass can solve this by returning both the maximum depth found in a subtree and the LCA of all deepest nodes within it. 3. Define the base case: If a node is null, return depth zero and null; if it is a leaf, return depth one and itself. 4. Execute the merge logic: Recursively call left and right children. Compare their returned depths. If equal, the current node is the LCA because deepest nodes exist in both subtrees. If unequal, propagate the deeper side up. 5. Optimize complexity: Emphasize that this approach runs in O(N) time and O(H) space, avoiding the brute-force method of calculating depths for every node separately.
Key Points to Cover
- Identify that a single DFS pass is sufficient rather than multiple traversals
- Correctly implement post-order logic where parent decisions depend on child results
- Handle the specific condition where left and right depths are equal to find the LCA
- Explain the O(N) time complexity advantage over naive approaches
- Demonstrate clear communication of the recursive state being returned
Sample Answer
To solve the Smallest Subtree with all the Deepest Nodes problem, I would use a post-order depth-first search approach. This allows us to determine the answer in a single pass through the tree. My strategy involves defin…
Common Mistakes to Avoid
- Calculating depth for every node individually, leading to inefficient O(N^2) performance
- Forgetting to return both the depth and the node reference from the recursive function
- Incorrectly handling the case where left and right subtree depths are equal
- Confusing the depth of the node with the depth of the subtree rooted at that node
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