LeetCode

Unique Binary Search Trees

2013.12.31 18:37

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:

　　See the definition for Catalan Number here, and that's the answer for this problem.

　　So, this problem is just to test if you know the term Catalan Number. If you do know the math formula, you won't use recurrence relation to calculate it, but use combinatorial number as a faster solution.

　　Time complexity is O(n), as calculating C(n, k) requires O(k) complexity, and F(n) = C(2n, n) / (n + 1).

　　Space complexity is O(1).

Accepted code:

class Solution {
public:
    int numTrees(int n) {
        // Note: The Solution object is instantiated only once and is reused by each test case.
        // res = C(2 * n, n) / (n + 1)
        return combination(2 * n, n) / (n + 1);
    }
private:
    int combination(int n, int k) {
        int i;
        int res, quo;
        
        if(k > n / 2){
            return combination(n, n - k);
        }

        res = quo = 1;
        for(i = 1; i <= k; ++i){
            res *= (n + 1 - i);
            quo *= i;
            if(res % quo == 0){
                res /= quo;
                quo = 1;
            }
        }

        return res;
    }
};