Introduction

In this blog post, we will explore how to solve the LeetCode problem "96" using AI. LeetCode is a popular platform for preparing for coding interviews, and with the help of AI tools like Chatmagic, we can generate solutions quickly and efficiently - helping you pass the interviews and get the job offer without having to study for months.

Problem Statement

Given an integer n, return the number of structurally unique BST's (binary search trees) which has exactly n nodes of unique values from 1 to n. Example 1: Input: n = 3 Output: 5 Example 2: Input: n = 1 Output: 1 Constraints: 1 <= n <= 19

Explanation

Here's the solution to the problem:

• Catalan Numbers: The number of structurally unique BSTs with n nodes is given by the nth Catalan number. The Catalan numbers can be computed using the formula: C(n) = (2n)! / ((n+1)! n!) or recursively C(n) = sum(C(i) C(n-1-i)) for i in range(n).
• Dynamic Programming: To avoid redundant calculations, we use dynamic programming to store and reuse the Catalan numbers for smaller values. Specifically, we will use the recursive approach in a DP manner.

• Base Cases: C(0) = 1 and C(1) = 1 are the base cases for the recursion.

• Runtime Complexity: O(n^2), Storage Complexity: O(n)

Code

    def numTrees(n: int) -> int:
    """
    Calculates the number of structurally unique BSTs with n nodes.

    Args:
        n: The number of nodes in the BST.

    Returns:
        The number of structurally unique BSTs.
    """

    dp = [0] * (n + 1)
    dp[0] = 1
    dp[1] = 1

    for i in range(2, n + 1):
        for j in range(i):
            dp[i] += dp[j] * dp[i - 1 - j]

    return dp[n]