Problem D

How Many Trees?

Input: standard input

Output: standard output

Memory Limit: 32 MB

 

A binary search tree is a binary tree with root k such that any node v in the left subtree of k has label (v) <label (k) and any node w in the right subtree of k has label (w) > label (k).

When using binary search trees, one can easily look for a node with a given label x: After we compare x to the label of the root, either we found the node we seek or we know which subtree it is in. For most binary search trees the average time to find one of its n nodes in this way is O(log n).

Given a number n, can you tell how many different binary search trees may be constructed with a set of numbers of size n such that each element of the set will be associated to the label of exactly one node in a binary search tree?

Input and Output

The input will contain a number 1 <= i <= 1000 per line representing the number of elements of the set. You have to print a line in the output for each entry with the answer to the previous question.

Sample Input

1

2

3

 

Sample Output

1

2

5

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公式 f[n] = C(2*n,n)/(n+1)
原本的遞迴公式 f[n]=f[0]*f[n-1]+f[1]*f[n-2]+…+f[n-1]*f[0]
產生遞迴後 f[n]=(4n-2)/(n+1)*f[n-1]


import java.math.BigInteger;
import java.util.Scanner;

public class Main {
    public static void main(String[] args) {
        BigInteger[] f = new BigInteger[1001];
        f[1] = BigInteger.valueOf(1);
        for(int i = 2; i <= 1000; i++) {
            f[i] = f[i-1].multiply(BigInteger.valueOf(4*i-2)).divide(BigInteger.valueOf(i+1));
        }
        Scanner cin = new Scanner(System.in);
        while(cin.hasNext()) {
            int n = cin.nextInt();
            System.out.println(f[n]);
        }
    }
}