LeetCode Unique Binary Search Trees

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Unique Binary Search Trees

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

动态规划法

本题关键：如何填表，如何得到计算公式.

需要很仔细画图，一步一步找出规律。

Catlan公式是可以计算这样的题目的，不过Catalan公式也不好理解，还是写个循环解决吧。

class Solution {public:int numTrees(int n){vector<int> ta(n+1);ta[0] = 1; ta[1] = 1; ta[2] = 2;for (int i = 3; i <= n; i++){int mid = (i-1)/2;for (int j = 0; j < mid; j++){ta[i] += ta[j] * ta[i-j-1] *2;}if (i%2) ta[i] += ta[mid] * ta[mid];else ta[i] += ta[mid+1] * ta[mid] * 2;}return ta[n];}};

//2014-2-15 updateint numTrees(int n){if (n < 3) return n;int *A = new int[n+1];A[0] = 1, A[1] = 1, A[2] = 2;for (int i = 3; i <= n; i++){int half = i/2;A[i] = 0;//注意要初始化for (int j = 1; j <= half; j++)A[i] += A[j-1] * A[i-j] *2;//注意细节是+=if (i%2 == 1) A[i] += A[half]*A[half];}return A[n];}//2014-2-15 update 2int numTrees2(int n){int *A = new int[n+1];A[0] = 1;for (int i = 1; i <= n; i++){A[i] = 0;//注意要初始化for (int j = 1; j <= i; j++)A[i] += A[j-1] * A[i-j];//注意细节是+=}return A[n];}

下面是参考资料：

http://zh.wikipedia.org/wiki/%E5%8D%A1%E5%A1%94%E5%85%B0%E6%95%B0

http://en.wikipedia.org/wiki/Catalan_number

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