Welcome to our comprehensive guide on solving LeetCode Problem 938: Range Sum of BST. In this video, we will delve into the details of this popular binary search tree (BST) problem, explore various solution strategies, and provide insights into how to optimize your approach for better performance.
Introduction
LeetCode Problem 938, titled "Range Sum of BST," is a fundamental problem that tests your understanding of binary search trees and range queries. The task is to calculate the sum of all node values in a BST that fall within a given range [L, R]. This problem is an excellent exercise for practicing tree traversal techniques and understanding how to leverage the properties of a BST to achieve an efficient solution.
Problem Statement
Given the root of a binary search tree and two integers L and R, return the sum of values of all nodes with a value in the inclusive range [L, R]. The BST is defined as follows:
Each node has at most two children.
The left subtree of a node contains only nodes with keys less than the node's key.
The right subtree of a node contains only nodes with keys greater than the node's key.
Example
Let's consider an example to better understand the problem:
Input:
makefile
Copy code
10
/ \
5 15
/ \ \
3 7 18
L = 7, R = 15
Output: 32
Explanation: Nodes within range [7, 15] are 7, 10, and 15. Their sum is 32.
Approach to the Solution
To solve this problem, we need to traverse the BST and accumulate the sum of node values that lie within the given range. There are several traversal techniques that we can use, but given the properties of a BST, certain methods are more efficient.
In-Order Traversal
One way to solve this problem is by performing an in-order traversal of the tree. In-order traversal visits nodes in ascending order, which makes it easy to check if a node's value falls within the range [L, R]. However, this approach might not be the most efficient because it does not take full advantage of the BST properties to prune unnecessary branches.
Depth-First Search (DFS)
A more optimized approach is to use DFS with pruning. The idea is to traverse the tree, but skip entire subtrees that we know do not contain any values within the range [L, R]. For example, if the current node's value is less than L, then we know that all nodes in the left subtree are also less than L and can be ignored. Similarly, if the current node's value is greater than R, then all nodes in the right subtree can be ignored.
Breadth-First Search (BFS)
Another approach is to use BFS, which involves level-order traversal. This method is also effective and can be implemented iteratively using a queue. Like the DFS approach, BFS can also be optimized by pruning subtrees that do not fall within the range [L, R].
Optimal Solution Strategy
The most efficient solution combines DFS with pruning. By leveraging the BST properties, we can skip entire subtrees and reduce the number of nodes we need to visit. Here's a step-by-step explanation of the optimal approach:
Start at the root node: Initialize the sum to 0.
Traverse the tree: Use a stack to implement iterative DFS.
Check node values: For each node, check if its value lies within the range [L, R].
If the node's value is within the range, add it to the sum.
If the node's value is greater than L, traverse the left subtree.
If the node's value is less than R, traverse the right subtree.
Prune subtrees: Skip subtrees that do not contain any values within the range.
Time and Space Complexity
The time complexity of this approach is O(N) in the worst case, where N is the number of nodes in the tree. This happens when all nodes fall within the range [L, R]. However, with effective pruning, the actual number of nodes visited can be significantly less. The space complexity is O(H), where H is the height of the tree, due to the stack used for the DFS traversal.
Conclusion
In this video, we have explored LeetCode Problem 938: Range Sum of BST in detail. We discussed different traversal techniques and identified the optimal solution strategy that leverages the properties of a binary search tree to efficiently calculate the sum of node values within a given range. By using DFS with pruning, we can achieve a time-efficient solution that minimizes the number of nodes visited.
Additional Tips
Understanding the properties of a binary search tree is crucial for solving this problem efficiently.
Practicing different tree traversal techniques will help you become more comfortable with similar problems.
Always consider edge cases, such as an empty tree or a range that does not include any node values.
Further Reading
For more information and practice problems on binary search trees, consider exploring the following resources:
"Introduction to Algorithms" by Cormen, Leiserson, Rivest, and Stein