What is the time complexity of Smallest Subtree with all the Deepest Nodes?

The optimal solution has a time complexity of O(n).

What data structures are used to solve Smallest Subtree with all the Deepest Nodes?

This problem uses Hash Table, Tree, Depth-First Search, Breadth-First Search, Binary Tree. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Smallest Subtree with all the Deepest Nodes?

Insight 1: We need to find the depth of the tree and the deepest nodes simultaneously. Insight 2: A recursive Depth-First Search (DFS) is a suitable approach to traverse the tree and compute depths.

Which companies ask LeetCode 865. Smallest Subtree with all the Deepest Nodes in interviews?

This problem is frequently asked at Salesforce.

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Smallest Subtree with all the Deepest Nodes - LeetCode 865 Solution

Smallest Subtree with all the Deepest Nodes - Complete Solution Guide

Smallest Subtree with all the Deepest Nodes is LeetCode problem 865, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

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. Example 1: Input: root = [3,5,1,6,2,0,8,null,null,7,4] Output: [2,7,4] Explanation: We return the node with

Detailed Explanation

The problem asks us to find the smallest subtree of a given binary tree that contains all the deepest nodes. A deepest node is a node with the maximum depth in the tree (i.e., the longest path from the root to that node). The subtree of a node consists of the node itself and all its descendants. We need to return the root of the smallest such subtree. Smallest here means the subtree whose root is closest to the deepest nodes.

Solution Approach

The solution uses a recursive Depth-First Search (DFS) approach to traverse the binary tree. During the traversal, for each node, we calculate the depth of its left and right subtrees. Based on the depths, we determine the Lowest Common Ancestor (LCA) of the deepest nodes. If the depths of the left and right subtrees are equal, the current node is the LCA. If one subtree is deeper than the other, the LCA is the LCA of the deeper subtree.

Step-by-Step Algorithm

Step 1: Define a recursive function `dfs(node)` that returns a tuple or object containing the depth of the subtree rooted at `node` and the LCA of the deepest nodes in that subtree.
Step 2: Base case: If `node` is null, return a depth of 0 and a null LCA.
Step 3: Recursively call `dfs` on the left and right children of `node` to obtain their respective depths and LCAs.
Step 4: Compare the depths of the left and right subtrees.
Step 5: If the left depth is greater than the right depth, the depth of the current subtree is `left_depth + 1`, and the LCA is the LCA of the left subtree.
Step 6: If the right depth is greater than the left depth, the depth of the current subtree is `right_depth + 1`, and the LCA is the LCA of the right subtree.
Step 7: If the left and right depths are equal, the depth of the current subtree is `left_depth + 1` (or `right_depth + 1`), and the LCA is the current node `node` itself.
Step 8: Return the depth and LCA from the `dfs` function.
Step 9: Call `dfs` on the root of the tree and return the LCA from the result.

Key Insights

Insight 1: We need to find the depth of the tree and the deepest nodes simultaneously.
Insight 2: A recursive Depth-First Search (DFS) is a suitable approach to traverse the tree and compute depths.
Insight 3: The Lowest Common Ancestor (LCA) of the deepest nodes is the root of the smallest subtree containing all deepest nodes.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(n)

Topics

This problem involves: Hash Table, Tree, Depth-First Search, Breadth-First Search, Binary Tree.

Companies

Asked at: Salesforce.

Smallest Subtree with all the Deepest Nodes | LeetCode 865

Medium

Hash Table Tree Depth-First Search Breadth-First Search Binary Tree

Time: O(n)Space: O(n)

Solution Code

Available in:

/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     int val;
 *     TreeNode left;
 *     TreeNode right;
 *     TreeNode() {}
 *     TreeNode(int val) { this.val = val; }
 *     TreeNode(int val, TreeNode left, TreeNode right) {
 *         this.val = val;
 *         this.left = left;
 *         this.right = right;
 *     }
 * }
 */
class Solution {
    public TreeNode subtreeWithAllDeepest(TreeNode root) {
        return dfs(root).lca;
    }

    private Pair dfs(TreeNode node) {
        if (node == null) {
            return new Pair(0, null);
        }

        Pair left = dfs(node.left);
        Pair right = dfs(node.right);

        if (left.depth > right.depth) {
            return new Pair(left.depth + 1, left.lca);
        } else if (right.depth > left.depth) {
            return new Pair(right.depth + 1, right.lca);
        } else {
            return new Pair(left.depth + 1, node);
        }
    }

    private class Pair {
        int depth;
        TreeNode lca;

        public Pair(int depth, TreeNode lca) {
            this.depth = depth;
            this.lca = lca;
        }
    }
}

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Problem Info

Description

Problem 865: Smallest Subtree with all the Deepest Nodes - Detailed explanation and solution approaches.

Topics

Hash Table Tree Depth-First Search Breadth-First Search Binary Tree

Companies

Salesforce

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:865

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

9 steps

1
Step 1: Define a recursive function `dfs(node)` that returns a tuple or object containing the depth of the subtree rooted at `node` and the LCA of the deepest nodes in that subtree.
2
Step 2: Base case: If `node` is null, return a depth of 0 and a null LCA.
3
Step 3: Recursively call `dfs` on the left and right children of `node` to obtain their respective depths and LCAs.
4
Step 4: Compare the depths of the left and right subtrees.
5
Step 5: If the left depth is greater than the right depth, the depth of the current subtree is `left_depth + 1`, and the LCA is the LCA of the left subtree.
6
Step 6: If the right depth is greater than the left depth, the depth of the current subtree is `right_depth + 1`, and the LCA is the LCA of the right subtree.
7
Step 7: If the left and right depths are equal, the depth of the current subtree is `left_depth + 1` (or `right_depth + 1`), and the LCA is the current node `node` itself.
8
Step 8: Return the depth and LCA from the `dfs` function.
9
Step 9: Call `dfs` on the root of the tree and return the LCA from the result.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n)

👍 Linear - Good! Visits each element once.

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The time complexity is determined by the number of nodes visited, which is proportional to n. The space complexity is determined by the maximum depth of the recursion stack, which in the worst case is n.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Level-Order Traversal (BFS)

Perform a level-order traversal to find the deepest level. Then, find the lowest common ancestor of all nodes at that deepest level.

Time: O(n)Space: O(w), where w is the maximum width of the tree. In the worst case, w can be n/2 for a complete binary tree.

Trade-offs: BFS requires more space than DFS (especially for wide trees). DFS is generally preferred because it can be implemented with similar time complexity but better space complexity in many cases.

✨

Key Insights

1Insight 1: We need to find the depth of the tree and the deepest nodes simultaneously.
2Insight 2: A recursive Depth-First Search (DFS) is a suitable approach to traverse the tree and compute depths.
3Insight 3: The Lowest Common Ancestor (LCA) of the deepest nodes is the root of the smallest subtree containing all deepest nodes.

📚

Key Terms Explained

For Students

Hash Table

A data structure that maps keys to values for quick lookup. Like a dictionary!

Tree

Hierarchical structure with nodes. Like a family tree!

Edge Cases

Edge case 1: Empty tree (root is null). The solution correctly handles this by returning null, as the depth and LCA are defined as 0 and null respectively in this scenario.
Edge case 2: Tree with only one node. The solution correctly identifies the root node as both the deepest node and the smallest subtree containing all deepest nodes.
Edge case 3: A complete binary tree where all leaf nodes are deepest. The algorithm correctly identifies the lowest common ancestor of the deepest leaf nodes, which can be the root or any intermediate node closer to the leaves.
Edge case 4: A skewed tree where all nodes are in a single chain. The solution properly identifies the deepest node as the smallest subtree containing all deepest nodes.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Incorrectly calculating the depth of a subtree. The depth should be calculated recursively by adding 1 to the maximum depth of the left or right subtree.

✗Common mistake 2: Failing to handle the case where the left and right subtrees have equal depths. In this case, the current node is the LCA.

✗Common mistake 3: Not considering the case when a node itself is the deepest node.

✗Common mistake 4: Assuming the binary tree is balanced which isn't always the case. The recursion depth will depend on the height of the tree, which can be O(n) in the worst case.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: In organizational charts, finding the smallest department containing all employees at the lowest level (e.g., most junior employees).

🏦Real-world scenario 2: In file systems, identifying the smallest directory containing all files that are at the maximum depth within a given directory structure.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

Hash tables trade space for time. Before using one, consider if the problem constraints allow the extra memory. In interviews, always mention this trade-off.

🎯 Interview Tip

If stuck, try working through a small example by hand. The pattern often reveals itself.

⚠️ Watch Out

Don't forget to handle key collisions and null/undefined keys. Consider what happens with duplicate values.

👨‍💻 From Experience

This problem pattern appears frequently in real codebases. I've used similar logic when building production systems.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: October 24, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 13

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 6

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJun 29

❓

Frequently Asked Questions

How do I choose between a hash map and a hash set?

Use a hash set when you only need to track presence/absence of elements (like detecting duplicates). Use a hash map when you need to associate values with keys (like counting frequencies or storing indices). Hash sets use less memory since they only store keys.

What is the difference between DFS and BFS for trees?

DFS (using recursion or stack) explores depth-first, useful for path-finding and backtracking problems. BFS (using queue) explores level-by-level, ideal for finding shortest paths or processing nodes by depth. Choose based on whether level information matters.

How much time should I spend on a medium problem in interviews?

In a 45-minute interview, aim for: 5-10 minutes understanding and clarifying, 5-10 minutes discussing approach, 20-25 minutes coding, and 5-10 minutes testing. If stuck after 15 minutes of thinking, verbalize your thought process and ask for hints.

What is the importance of time and space complexity analysis?

Complexity analysis is crucial because: 1) Interviewers always ask about it, 2) It helps you choose between approaches, 3) It demonstrates CS fundamentals. Always state both time and space complexity, and be prepared to explain how you derived them.

What is the optimal approach for "Smallest Subtree with all the Deepest Nodes"?

For this medium-level Hash Table problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether tree techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

🤖 Need help with this problem?