To balance a binary search tree, the nodes must be reordered using an in-order traversal to create a sorted list. Then, reconstruct a balanced tree by picking the middle element recursively as the root. This method ensures a balanced tree with logarithmic depth differences between subtrees.
Problem Statement
Given the root of a binary search tree, your task is to return a balanced binary search tree using the same node values. A balanced binary search tree has subtrees whose depths do not differ by more than one. If there are multiple correct answers, any valid solution will be accepted.
The problem requires converting the binary search tree into a sorted array by performing an in-order traversal. Once sorted, the tree should be reconstructed by selecting the middle node as the root and recursively balancing the left and right subtrees.
Examples
Example 1
Input: root = [1,null,2,null,3,null,4,null,null]
Output: [2,1,3,null,null,null,4]
This is not the only correct answer, [3,1,4,null,2] is also correct.
Example 2
Input: root = [2,1,3]
Output: [2,1,3]
Example details omitted.
Constraints
- The number of nodes in the tree is in the range [1, 104].
- 1 <= Node.val <= 105
Solution Approach
In-order Traversal
The first step is to traverse the tree in an in-order fashion, which gives us a sorted array of all the node values. This ensures that we can easily access the middle element to serve as the root for each subtree.
Recursive Tree Reconstruction
Once we have the sorted array, we recursively pick the middle element as the root and build the left and right subtrees using the same logic. This guarantees that the resulting tree is balanced.
Optimizing Space Complexity
To optimize space complexity, consider using a tree traversal that does not require additional arrays, or implement the in-order traversal iteratively. This can reduce the space complexity from O(n) to O(log n) for stack usage.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | O(n) |
| Space | O(n) |
The time complexity is O(n) because we perform an in-order traversal of the tree and rebuild the tree recursively, both of which require processing every node once. The space complexity is O(n) due to the space required for storing the sorted array and recursive calls in the stack.
What Interviewers Usually Probe
- Candidate should demonstrate understanding of binary search tree properties.
- Look for an efficient use of recursion in constructing the tree.
- Pay attention to the explanation of how to balance the tree using the middle element.
Common Pitfalls or Variants
Common pitfalls
- Not using in-order traversal to get the sorted node values.
- Incorrectly choosing the root node (must be the middle of the sorted list).
- Failing to handle large trees efficiently, leading to stack overflows or memory issues.
Follow-up variants
- How would this approach change if the input was a linked list?
- What if the tree is already balanced? Can we optimize the process?
- What if we need to maintain the original structure of the tree instead of balancing it?
How GhostInterview Helps
- GhostInterview guides you through the step-by-step process of transforming the tree, ensuring an efficient approach to balancing it.
- It highlights the most common issues, like incorrect tree traversal and improper node selection, to help you avoid pitfalls.
- The platform offers sample tests and real-time feedback on the recursive approach to optimize the solution.
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 main pattern in the 'Balance a Binary Search Tree' problem?
The key pattern is binary-tree traversal combined with state tracking. In this problem, the tree is traversed in-order to create a sorted array, and then the tree is reconstructed in a balanced way by selecting the middle element.
How do I balance a binary search tree efficiently?
The efficient way is by performing an in-order traversal to collect node values in a sorted array, then recursively constructing the tree by selecting the middle element of the array as the root.
Can multiple answers exist for this problem?
Yes, there can be multiple valid solutions as long as the tree is balanced. The exact structure may vary, but the general tree properties should remain the same.
What is the time complexity of the solution?
The time complexity is O(n), where n is the number of nodes in the tree, because each node is visited once during the in-order traversal and the tree reconstruction.
Why is space complexity O(n)?
Space complexity is O(n) due to the space needed for the sorted array (storing node values) and the recursive stack used during tree reconstruction.
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Find the maximum sum of values from any Binary Search Tree (BST) subtree in a binary tree.
Open problem page#1305 All Elements in Two Binary Search TreesMerge elements from two binary search trees and return them sorted in ascending order.
Open problem page#1569 Number of Ways to Reorder Array to Get Same BSTDetermine the number of ways to reorder an array to get the same binary search tree (BST) from its insertion order.
Open problem page