To solve the Smallest Subtree with all the Deepest Nodes problem, we need to find the smallest subtree that contains all the deepest nodes of the binary tree. By performing a depth-first search (DFS), we calculate the depth of each node and identify the deepest nodes. We then trace back to the common ancestor of the deepest nodes, which is the root of the smallest subtree containing them.
Problem Statement
Given the root of a binary tree, where each node's depth is the shortest distance to the root, you are tasked with finding the smallest subtree that contains all the deepest nodes in the tree. A node is considered the deepest if it has the largest depth possible among all the nodes in the tree.
Your goal is to return the root of this smallest subtree that encompasses all the deepest nodes. In case of multiple candidates, you should return the subtree with the smallest size.
Examples
Example 1
Input: root = [3,5,1,6,2,0,8,null,null,7,4]
Output: [2,7,4]
We return the node with value 2, colored in yellow in the diagram. The nodes coloured in blue are the deepest nodes of the tree. Notice that nodes 5, 3 and 2 contain the deepest nodes in the tree but node 2 is the smallest subtree among them, so we return it.
Example 2
Input: root = [1]
Output: [1]
The root is the deepest node in the tree.
Example 3
Input: root = [0,1,3,null,2]
Output: [2]
The deepest node in the tree is 2, the valid subtrees are the subtrees of nodes 2, 1 and 0 but the subtree of node 2 is the smallest.
Constraints
- The number of nodes in the tree will be in the range [1, 500].
- 0 <= Node.val <= 500
- The values of the nodes in the tree are unique.
Solution Approach
Depth-First Search (DFS)
Perform a DFS traversal starting from the root of the tree. For each node, recursively explore its left and right subtrees, calculating the depth and determining the deepest nodes. As you explore, track the deepest level and identify all nodes that match this level.
Tracking the Subtree Containing Deepest Nodes
After identifying the deepest nodes, trace back through the tree to find the lowest common ancestor of these nodes. This ancestor is the root of the smallest subtree that includes all deepest nodes. This requires comparing the depths of the left and right subtrees of each node.
Optimizing with State Tracking
Optimize the DFS approach by storing the depths and the subtree for each node. By checking the depth of left and right children, you can efficiently identify the smallest subtree containing the deepest nodes, ensuring that you don't redo calculations unnecessarily.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | O(N) |
| Space | O(N) |
The time complexity of the solution is O(N), where N is the number of nodes in the tree. This is because we visit each node once during the DFS traversal. The space complexity is also O(N) due to the space used by the recursion stack and the storage of subtree information for each node.
What Interviewers Usually Probe
- Test the candidate's ability to implement tree traversal algorithms like DFS.
- Evaluate how well the candidate optimizes space usage during the process.
- Check if the candidate correctly identifies the concept of common ancestors and subtree size.
Common Pitfalls or Variants
Common pitfalls
- Failing to correctly track the depth of nodes, leading to incorrect identification of the deepest nodes.
- Confusing the concept of subtree size with just the depth of nodes.
- Not properly handling edge cases like a tree with a single node or nodes with equal depths.
Follow-up variants
- Finding the subtree for the deepest node given a weighted binary tree.
- Handling unbalanced trees with varying depths across subtrees.
- Modifying the problem to identify the smallest subtree containing a specific set of nodes.
How GhostInterview Helps
- GhostInterview guides you through solving the problem step-by-step, ensuring you focus on binary-tree traversal and state tracking.
- The tool helps optimize your solution for time and space complexity, providing tips for effective DFS implementations.
- It simulates interview conditions by offering hints on how to discuss and optimize your approach in real-time.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the time complexity of the Smallest Subtree with all the Deepest Nodes problem?
The time complexity is O(N), where N is the number of nodes in the tree, as we perform a DFS traversal of the entire tree.
How do I find the smallest subtree with the deepest nodes in a binary tree?
You can find the smallest subtree by performing a DFS, calculating the depths of nodes, and identifying the lowest common ancestor of the deepest nodes.
What is the significance of tracking the depth of each node in this problem?
Tracking the depth allows you to identify the deepest nodes and their common ancestor, which is crucial for finding the smallest subtree.
Are there any edge cases I should consider when solving this problem?
Yes, edge cases include trees with a single node, trees with equal depth nodes, or unbalanced trees where depth varies significantly across subtrees.
Can I optimize the space complexity of my solution?
Yes, by tracking subtree information efficiently and avoiding redundant calculations, you can optimize space usage in your DFS traversal.
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