96 Unique Binary Search Trees

96. Unique Binary Search Trees

1. Question

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

Example:

Input:
3

Output:
5

Explanation:

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

2. Implementation

(1) DP

思路: Catalan数公式

class Solution {
    public int numTrees(int n) {
        if (n < 0) {
            return 0;
        }

        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 - j - 1];
            }
        }
        return dp[n];
    }
}

3. Time & Space Complexity

DP: 时间复杂度O(n), 空间复杂度O(n)