64-Unique Binary Search Trees

96. Unique Binary Search Trees

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Question
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Total Accepted: 82788 Total
Submissions: 220611 Difficulty: Medium

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

思路：  1 2 3 4 5 | 6 |  7 ..........n

既然是二叉搜索树，先递增排好序，然后在序列中选定一个根，比如上图的6

然后分为左子树和右子树，这样便可以转化为子问题

设f(n)为n个数的二叉排序树的数目

则f(n) = f(0)f(n-1)+f(1)f(n-2)+.......+f(n-1)f(0)

自底向上写出公式即可

时间复杂度O(n*n),空间O(1)

效率更高可考虑直接用卡特兰公式

class Solution {
public:
    int numTrees(int n) {
        vector<int> f(n+1);
        f[0]=1;
        for(int i=1;i<=n;++i)
        {
            for(int k=0;k<=i-1;++k)
                f[i]+=f[k]*f[i-1-k];
        }
        return f[n];
    }
};