Unique Binary Search Trees II
Given an integer $n$, return all structurally unique Binary Search Trees (BSTs) which store values 1 to $n$. Use Recursion/DP.
Why Interviewers Ask This
Meta evaluates this question to assess a candidate's mastery of recursive backtracking and dynamic programming optimization. They specifically look for the ability to handle state explosion in combinatorial problems while maintaining clean, readable code. The problem tests if you can decompose a complex structural generation task into smaller subproblems involving root selection and left/right subtree construction.
How to Answer This Question
1. Clarify the problem constraints and confirm that values 1 to n must form a valid BST where inorder traversal yields sorted order. 2. Propose a recursive strategy: iterate through each number i from start to end, treating it as the root. 3. Explain how to generate all possible left subtrees using numbers less than i and all right subtrees using numbers greater than i. 4. Combine every unique left subtree with every unique right subtree to form complete trees rooted at i. 5. Discuss memoization (DP) to cache results for ranges [start, end], preventing redundant calculations for identical sub-ranges. 6. Conclude by analyzing time complexity O(4^n / sqrt(n)) and space complexity, showing awareness of Meta's focus on efficient algorithmic design.
Key Points to Cover
- Explicitly stating the relationship between the chosen root and the partitioning of remaining values into left and right sets
- Demonstrating the Cartesian product logic for combining independent left and right subtree lists
- Implementing memoization to avoid recalculating identical sub-range results
- Correctly handling base cases where the start index exceeds the end index
- Articulating the exponential time complexity related to Catalan numbers
Sample Answer
To solve Unique Binary Search Trees II, I first recognize that for any range of numbers, choosing a specific number as the root dictates the structure of its left and right children based on BST properties. My approach u…
Common Mistakes to Avoid
- Failing to handle the empty range case correctly, leading to null pointer exceptions when building leaf nodes
- Not using memoization, resulting in Time Limit Exceeded errors due to exponential recomputation
- Confusing value uniqueness with structural uniqueness, potentially generating duplicate tree structures
- Ignoring the BST property that left children must be smaller and right children larger than the root
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