Leader in Tree Land

Time Limit: 2000/1000 MS (Java/Others)    Memory
Limit: 65536/65536 K (Java/Others)

Problem Description

Tree land has  cities,
connected by  roads.
You can go to any city from any city. In other words, this land is a tree. The city numbered one is the root of this tree.

There are  ministers
numbered from  to .
You will send them to  cities,
one city with one minister.

Since this is a rooted tree, each city is a root of a subtree and there are  subtrees.
The leader of a subtree is the minister with maximal number in this subtree. As you can see, one minister can be the leader of several subtrees.

One day all the leaders attend a meet, you find that there are exactly  ministers.
You want to know how many ways to send  ministers
to each city so that there are  ministers
attend the meet.

Give your answer mod .

Input

Multiple test cases. In the first line there is an integer ,
indicating the number of test cases. For each test case, first line contains two numbers .
Next  line
describe the roads of tree land.

Output

For each test case, output one line. The output format is Case #: ,  is
the case number,starting from .

Sample Input

2
3 2
1 2
1 3
10 8
2 1
3 2
4 1
5 3
6 1
7 3
8 7
9 7
10 6

Sample Output

Case #1: 4
Case #2: 316512

Source

2015 Multi-University Training Contest 7

解题思路：

可以用求概率的思想来解决这个问题。令以i号节点为根的子树为第i棵子树，设这颗子树恰好有sz[i]个点。那么第i个点是第i棵子树最大值的概率为1/sz[i]，不是最大值的概率为(sz[i]-1)/sz[i]。现在可以求解恰好有k个最大值的概率。

令dp[i][j]表示考虑编号从1到i的点，其中恰好有j个点是其子树最大值的概率。 很容易得到如下转移方程：dp[i][j]=dp[i-1][j]*(sz[i]-1)/sz[i]+dp[i-1][j-1]/sz[i]。这样dp[n][k]就是所有点中恰好有k个最大值的概率。

题目要求的是方案数，用总数n！乘上概率就是答案。计算的时候用逆元代替上面的分数即可

#include <iostream>
#include <cstring>
#include <cstdlib>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <vector>
#include <queue>
#include <stack>
#include <set>
#define LL long long
using namespace std;
const int MAXN = 1000 + 10;
const int MOD = 1000000000 + 7;
int read()
{
    int res = 0, f = 1; char ch = getchar();
    while(ch < '0' || ch > '9'){if(ch == '-') f *= -1; ch = getchar();}
    while(ch >= '0' && ch <= '9'){res = res * 10 + ch - '0'; ch = getchar();}
    return res;
}
int sz[MAXN], vis[MAXN];
int n, k;
LL dp[MAXN][MAXN], inv[MAXN], fac[MAXN];
struct Edge
{
    int to, next;
}edge[MAXN<<1];
int tot, head[MAXN];
void init()
{
    tot = 0;
    for(int i=0;i<=n;i++)
    {
        head[i] = -1;
        vis[i] = 0;
        sz[i] = 0;
    }
}
void addedge(int u, int v)
{
    edge[tot].to = v;
    edge[tot].next = head[u];
    head[u] = tot++;
}
int dfs(int u, int pre)
{
    int ans = 1;vis[u] = 1;
    for(int i=head[u];i!=-1;i=edge[i].next)
    {
        int v = edge[i].to;
        if(v == pre) continue;
        if(!vis[v]) ans += dfs(v, u);
    }
    return sz[u] = ans;
}
LL pow_mod(LL a, LL b)
{
    LL res = 1;
    while(b)
    {
        if(b & 1) res = res * a % MOD;
        a = a * a % MOD;
        b >>= 1;
    }
    return res % MOD;
}
int main()
{
    int T, kcase = 1;
    for(int i=1;i<MAXN;i++) inv[i] = pow_mod(i, MOD - 2);
    fac[0] = 1; for(int i=1;i<MAXN;i++) fac[i] = fac[i-1] * i % MOD;
    T = read();
    while(T--)
    {
        n = read(); k = read();
        int u, v; init();
        for(int i=1;i<n;i++)
        {
            u = read();v = read();
            addedge(u, v);
            addedge(v, u);
        }
        dfs(1, -1);
        for(int i=0;i<=n;i++) for(int j=0;j<=k;j++) dp[i][j] = 0;
        dp[0][0] = 1;
        for(int i=1;i<=n;i++)
        {
            for(int j=0;j<=k;j++)
            {
                dp[i][j] = ((dp[i-1][j] * (sz[i] - 1)) % MOD) * inv[sz[i]] % MOD +
                dp[i-1][j-1] * inv[sz[i]] % MOD;
            }
        }
        LL ans = (dp[n][k] * fac[n]) % MOD;
        printf("Case #%d: %I64d\n", kcase++, ans);
    }
    return 0;
}

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