# 96. Unique Binary Search Trees

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

# Solution

Approach 1: DP, e.g., use a table to prestore previously computed results. Time: O(N^2).

# Code (Python)

Approach 1:

    def numTrees(self, n: 'int') -> 'int':
        # naive recursion
        
        #if n == 0:
        #    return 1
        #if n == 1:
        #    return 1
        #total = 0
        #for i in range(n):
        #    total += self.numTrees(i) * self.numTrees(n - 1 - i)
        #return total
    
        # dp: O(N^2)
        nums = [0 for _ in range(n + 1)]
        nums[0], nums[1] = 1, 1
        for i in range(2, n + 1):
            for j in range(i):
                nums[i] += nums[j] * nums[i - 1 - j]
        return nums[-1]

# Code (C++)

Approach 1:

class Solution {
public:
    int numTrees(int n) {
        int treeNum[n+1] = {0};
        treeNum[0] = 1;
        for (int len = 1; len <= n; ++len)
        {
            for (int root = 1; root <= len; ++root)
            {
                int leftLen = root - 1;
                int rightLen = len - root;
                treeNum[len] += treeNum[leftLen] * treeNum[rightLen];
            }
        }
        return treeNum[n];
    }
};