leetcode96 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
"""
"""
数学分析，让每个值作为根结点，再讨论
左右子树结点的数量，能得到
res[1]=1
res[2]=res[1]*res[0]+res[0]*res[1]
res[3]=(res[0]*res[2])+(res[1]*res[1])+(res[2]*res[0])
令res[0]=1保证树的一边为空存在
"""
class Solution:
    def numTrees(self, n):
        res = [0]*(n+1)
        res[0], res[1] = 1, 1
        for i in range(2, n+1):
            for j in range(i):
                res[i] += res[j]*res[i-j-1]
        return res[n]