96. Unique Binary Search Trees

Given n, how many structurally unique BST's (binary search trees) that store values 1...n?

For example,
Given n = 3, there are a total of 5 unique BST's.

   1         3     3      2      1
    \       /     /      / \      \
     3     2     1      1   3      2
    /     /       \                 \
   2     1         2                 3

Solution:

Use dynamic programming to solve this problem.

Firstly, given i nodes, if we put x nodes at left and y nodes at right based on BST rule, the problem becomes: number of combination with x nodes times number of combination with y nodes.

Therefore, we use

dp[i]  to denote the number of combinations with i nodes.

dp[0] = 1 since 0 node only has one combination.
dp[1] = 1. 1 node also has only one combination.

dp[i] = dp(0) * dp(i - 1) + dp(1) * dp(i - 2) + ... + dp(i - 2) * dp(1) + dp(i - 1) * dp(0)

= ∑ dp[0...k] * [ k+1....i]     0<=k<i-1



Code:

public class Solution {
    public int numTrees(int n) {
        int[] dp = new int[n + 1];
        dp[0] = 1;
        dp[1] = 1;
        for (int i = 2; i <= n; i++) {
            for (int j = 0; j < i; j++) {
                dp[i] += dp[j] * dp[i - 1 - j];
            }
        }
        return dp[n];
    }
}