human
Can you provide a Python program to print the expectation of the number of moves given the following:
A maze is represented by a tree (an undirected graph, where exactly one way exists between each pair of vertices). In the maze the entrance vertex and the exit vertex are chosen with some probability. The exit from the maze is sought by Deep First Search. If there are several possible ways to move, the move is chosen equiprobably. Consider the following pseudo-code:
    
    
      
    DFS(x)  
        if x == exit vertex then  
            finish search  
        flag[x] <- TRUE  
        random shuffle the vertices' order in V(x) // here all permutations have equal probability to be chosen  
        for i <- 1 to length[V] do  
            if flag[V[i]] = FALSE then  
                count++.
 
                DFS(y).
 
        count++.
 
    

V(x) is the list vertices adjacent to x. The flag array is initially filled as FALSE. DFS initially starts with a parameter of an entrance vertex. When the search is finished, variable count will contain the number of moves.

Your task is to count the mathematical expectation of the number of moves one has to do to exit the maze.

Input

The first line determines the number of vertices in the graph n (1 ≤ n ≤ 105). The next n - 1 lines contain pairs of integers ai and bi, which show the existence of an edge between ai and bi vertices (1 ≤ ai, bi ≤ n). It is guaranteed that the given graph is a tree.

Next n lines contain pairs of non-negative numbers xi and yi, which represent the probability of choosing the i-th vertex as an entrance and exit correspondingly. The probabilities to choose vertex i as an entrance and an exit equal  and  correspondingly. The sum of all xi and the sum of all yi are positive and do not exceed 106.

Output

Print the expectation of the number of moves. The absolute or relative error should not exceed 10 - 9.

Examples

Input

2
1 2
0 1
1 0

Output

1.00000000000000000000

Input

3
1 2
1 3
1 0
0 2
0 3

Output

2.00000000000000000000

Input

7
1 2
1 3
2 4
2 5
3 6
3 7
1 1
1 1
1 1
1 1
1 1
1 1
1...
impor
assistant
#Here is Python program to print the expectation of the number of moves.
t java.io.*;
import java.util.*;

public class Main{
	
	public int m,n,j,k,l,i,o,Be[],E[],N[],T,in[],ou[],Sin,Sout,sum[]; 
	public double ans;
	public static void main(String args[]) throws IOException{
		new Main().mains();
	}
	
	void make(int i, int j, int k){
		N[++ T] = Be[i]; Be[i] = T; E[T] = j; 
		if (k == 1) make(j,  i, 0);
	}
	
	public void solve() throws IOException{
		m = nextInt();
		Be = new int[m * 2 + 10];
		E = new int[m * 2 + 10];
		N = new int[m * 2 + 10];
		in = new int[m * 2 + 10];
		ou = new int[m * 2 + 10];
		sum = new int[m * 2 + 10];
		int stack[] = new int[m * 2 + 10];
		int alr[] = new int[m * 2 + 10];
		int top = 0;
		int now[] = new int[m * 2 + 10];
		
		for (int i = 1; i < m; i ++){
			j = nextInt();
			k = nextInt();
			make(j, k, 1);
		}
		for (int i = 1; i <= m; i ++){
			in[i] = nextInt();
			ou[i] = nextInt();
			Sin += in[i];
			Sout += ou[i];
		}
		stack[top = 1] = 1;
		now[1] = Be[1];
		sum[1] = 1;
		for (; top != 0; ){
			int No = stack[top];
			if (alr[No] == 1){
				int def = E[now[No]];
				sum[No] += sum[def];
				in[No] += in[def];
				ans += ((double)in[def]) * ou[No] * sum[def];
				alr[No] = 0;
				now[No] = N[now[No]];
				continue;
			}
			if (now[No] == 0){
				ans += ((double)(Sin - in[No])) * ou[No] * (m - sum[No]);
				top --;
				continue;
			}
			int def = E[now[No]];
			if (def == stack[top - 1]){
				now[No] = N[now[No]];
				continue;
			}
			stack[++ top] = def;
			now[def] = Be[def];
			sum[def] = 1;
			alr[No] = 1;
		}
		out.printf("%.10f\n", ans / ((double)Sin) / ((double)Sout));
	}
	
	void mains() throws IOException{
		reader = new BufferedReader(new InputStreamReader(System.in));
		out = new PrintWriter(new OutputStreamWriter(System.out));
		tokenizer = null;
		//File fin = new File("a.in");
		//reader = new BufferedReader(new InputStreamReader(new FileInputStream(fin)));
		//out = new PrintWriter("a.out");
		solve();
		reader.close();
		out.flush();
	}
	
	BufferedReader reader;
	StringTokenizer tokenizer;
	PrintWriter out;
	int nextInt() throws...