How to Solve Merge BSTs to Create Single BST

You are given an array of small BSTs. The task is to merge these BSTs into one valid BST using specific operations. If merging is impossible at any step, return null. The main challenge involves binary-tree traversal and managing state during merging.

Problem Statement

You are given an array of n BST root nodes. Each tree in the array has at most 3 nodes. The task is to merge these trees into a single valid BST using a series of operations. At each step, you can pick two trees, merge them into a valid BST, and remove the merged trees from the array. If at any point you cannot merge the trees into a valid BST, return null.

A valid BST must satisfy the binary search property: for each node, the values in the left subtree are smaller, and the values in the right subtree are larger. The goal is to determine if it's possible to perform n-1 operations to merge all the trees into one valid BST, or return null if such a sequence of operations is not possible.

Examples

Example 1

Input: trees = [[2,1],[3,2,5],[5,4]]

Output: [3,2,5,1,null,4]

In the first operation, pick i=1 and j=0, and merge trees[0] into trees[1]. Delete trees[0], so trees = [[3,2,5,1],[5,4]].

In the second operation, pick i=0 and j=1, and merge trees[1] into trees[0]. Delete trees[1], so trees = [[3,2,5,1,null,4]].

The resulting tree, shown above, is a valid BST, so return its root.

Example 2

Input: trees = [[5,3,8],[3,2,6]]

Output: []

Pick i=0 and j=1 and merge trees[1] into trees[0]. Delete trees[1], so trees = [[5,3,8,2,6]].

The resulting tree is shown above. This is the only valid operation that can be performed, but the resulting tree is not a valid BST, so return null.

Example 3

Input: trees = [[5,4],[3]]

Output: []

It is impossible to perform any operations.

Constraints

• n == trees.length
• 1 <= n <= 5 * 104
• The number of nodes in each tree is in the range [1, 3].
• Each node in the input may have children but no grandchildren.
• No two roots of trees have the same value.
• All the trees in the input are valid BSTs.
• 1 <= TreeNode.val <= 5 * 104.

Solution Approach

Binary Tree Traversal

To solve this, you will need to traverse the trees and check if merging two BSTs at each step maintains the BST properties. This can be done with an in-order traversal that ensures the left subtree has smaller values and the right subtree has larger values.

State Tracking During Merging

During each merge, you must track the state of the trees being merged, ensuring that each merge operation results in a valid BST. This includes comparing the values of the root nodes and their children to ensure the merging process does not violate the BST property.

Efficient Merging Strategy

An efficient approach would involve leveraging binary search properties for merging. After traversing the trees, it may help to track possible roots and merge in a manner that minimizes violations of the BST property.

Complexity Analysis

The time complexity of the solution depends on the approach you take for merging the trees, particularly whether you optimize for efficient tree traversal and merging. Space complexity will vary based on the space required for storing temporary trees and managing state during the merges.

What Interviewers Usually Probe

• Can the candidate manage binary tree traversal while ensuring the BST properties are preserved?
• Is the candidate able to track the state of multiple BSTs and efficiently decide which trees to merge?
• Does the candidate understand the implications of BST properties during tree merging?

Common Pitfalls or Variants

Common pitfalls

• Failing to check if a merged tree still satisfies the BST property after each merge.
• Merging two trees that result in an invalid BST, leading to an incorrect result.
• Not efficiently managing the array of trees to ensure optimal merges.

Follow-up variants

• What if the trees have different sizes or larger trees are allowed?
• What if the merging process should allow for more than n-1 operations?
• How would the solution change if some trees are already invalid?

How GhostInterview Helps

• GhostInterview guides you through tree traversal, ensuring your merges maintain the BST property throughout the solution.
• The platform helps optimize your merging strategy, ensuring you manage multiple trees and track the state effectively.
• GhostInterview's hints will allow you to visualize the tree merging process and avoid common pitfalls in ensuring a valid BST.

Topic Pages

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

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FAQ

What is the main challenge in the "Merge BSTs to Create Single BST" problem?

The main challenge is to merge multiple BSTs while maintaining the binary search tree properties. You must ensure each merge operation results in a valid BST.

How can I efficiently merge multiple BSTs?

You should leverage binary search properties and track the state of each merge, ensuring that every tree remains valid throughout the merging process.

What should I do if merging results in an invalid BST?

If a merge results in an invalid BST, return null, as it's not possible to create a valid BST through that sequence of operations.

How does GhostInterview assist with solving the "Merge BSTs to Create Single BST" problem?

GhostInterview provides guidance on binary tree traversal, state tracking during merges, and optimal merging strategies, helping you avoid common mistakes.

What are some common pitfalls when solving this problem?

Common pitfalls include failing to check the BST property after merging, inefficiently managing the tree array, and merging trees that result in an invalid BST.