Problem:
Score : 600 points

Problem Statement

Given is a tree T with N vertices. The i-th edge connects Vertex A_i and B_i (1 \leq A_i,B_i \leq N).

Now, each vertex is painted black with probability 1/2 and white with probability 1/2, which is chosen independently from other vertices. Then, let S be the smallest subtree (connected subgraph) of T containing all the vertices painted black. (If no vertex is painted black, S is the empty graph.)

Let the holeyness of S be the number of white vertices contained in S. Find the expected holeyness of S.

Since the answer is a rational number, we ask you to print it \bmod 10^9+7, as described in Notes.

Notes

When you print a rational number, first write it as a fraction \frac{y}{x}, where x, y are integers, and x is not divisible by 10^9 + 7
(under the constraints of the problem, such representation is always possible).

Then, you need to print the only integer z between 0 and 10^9 + 6, inclusive, that satisfies xz \equiv y \pmod{10^9 + 7}.

Constraints

2 \leq N \leq 2 \times 10^5

1 \leq A_i,B_i \leq N

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N
A_1 B_1
:
A_{N-1} B_{N-1}

Output

Print the expected holeyness of S, \bmod 10^9+7.

Sample Input 1

3
1 2
2 3

Sample Output 1

125000001

If the vertices 1, 2, 3 are painted black, white, black, respectively, the holeyness of S is 1.

Otherwise, the holeyness is 0, so the expected holeyness is 1/8.

Since 8 \times 125000001 \equiv 1 \pmod{10^9+7}, we should print 125000001.

Sample Input 2

4
1 2
2 3
3 4

Sample Output 2

375000003

The expected holeyness is 3/8.

Since 8 \times 375000003 \equiv 3 \pmod{10^9+7}, we should print 375000003.

Sample Input 3

4
1 2
1 3
1 4

Sample Output 3

250000002

The expected holeyness is 1/4.

Sample Input 4

7
4 7
3 1
2 6
5 2
7 1
2 7

Sample Output 4

570312505

Solution in C# (.NET Core 3.1.201) (judge verdict: Wrong Answer)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using static System.Console;
using static System.Math;
 
public class Solve{
    static public int mod = 1000000007;
    static public string al = "abcdefghijklmnopqrstuvwxyz";
    public static void Main(){
        // •ûj
        // なにもわからん
        var n = rint();
        var path = new List<List<int>>();
        for(int i=0;i<n;i++){
            path.Add(new List<int>());
        }
        for(int i=0;i<n-1;i++){
            var a = inta();
            a[0]--;
            a[1]--;
            path[a[0]].Add(a[1]);
            path[a[1]].Add(a[0]);
        }
        
        var llist = new long[n];
        var rlist = new long[n];
        
        var check = -1;
        long ki1c = 0;
        long ki2c = 0;
        long ki3c = 0;
        for(int i=0;i<n;i++){
            if(path[i].Count == 1) continue;
            if(path[i].Count == 3){
                var ki1 = path[i][0];
                var ki2 = path[i][1];
                var ki3 = path[i][2];
                var passedk = new bool[n];
                passedk[i] = true;
                
                var queuek = new Queue<int>();
                queuek.Enqueue(ki1);
                while(queuek.Count > 0){
                    var de = queuek.Dequeue();
                    if(passedk[de]) continue;
                    passedk[de] = true;
                    ki1c++;
                    for(int j=0;j<path[de].Count;j++){
                        queuek.Enqueue(path[de][j]);
                    }
                }
                queuek.Enqueue(ki2);
                while(queuek.Count > 0){
                    var de = queuek.Dequeue();
                    if(passedk[de]) continue;
                    passedk[de] = true;
                    ki2c++;
                    for(int j=0;j<path[de].Count;j++){
                        queuek.Enqueue(path[de][j]);
                    }
                }
                queuek.Enqueue(ki3);
                while(queuek.Count > 0){
                    var de = queuek.Dequeue();
                    if(passedk[de]) continue;
                    passedk[de] = true;
                    ki3c++;
                    for(int j=0;j<path[de].Count;j++){
                        queuek.Enqueue(path[de][j]);
                    }
                }
                
                check = i;
                continue;
            }
            var l = path[i][0];
            var r = path[i][1];
            var passed = new bool[n];
            passed[i] = true;
            
            var queue = new Queue<int>();
            queue.Enqueue(l);
            while(queue.Count > 0){
                var de = queue.Dequeue();
                if(passed[de]) continue;
                passed[de] = true;
                llist[i]++;
                for(int j=0;j<path[de].Count;j++){
                    queue.Enqueue(path[de][j]);
                }
            }
            queue.Enqueue(r);
            while(queue.Count > 0){
                var de = queue.Dequeue();
                if(passed[de]) continue;
                passed[de] = true;
                rlist[i]++;
                for(int j=0;j<path[de].Count;j++){
                    queue.Enqueue(path[de][j]);
                }
            }
        }
        
        long anaaki = 0;
        for(int i=0;i<n;i++){
            llist[i] = square(2,(int)llist[i])-1;
            rlist[i] = square(2,(int)rlist[i])-1;
            if(llist[i] < 0) llist[i] += mod;
            if(rlist[i] < 0) rlist[i] += mod;
            anaaki = (anaaki + llist[i]*rlist[i]) % mod;
        }
        ki1c = square(2,(int)ki1c)-1;
        ki2c = square(2,(int)ki2c)-1;
        ki3c = square(2,(int)ki3c)-1;
        if(ki1c < 0) ki1c += mod;
        if(ki2c < 0) ki2c += mod;
        if(ki3c < 0) ki3c += mod;
        anaaki = (anaaki + ki1c*ki2c) %mod;
        anaaki = (anaaki + ki3c*ki2c) %mod;
        anaaki = (anaaki + ki1c*ki3c) %mod;
        anaaki = (anaaki + ki1c*ki2c*ki3c) %mod;
        //WriteLine(anaaki);
        
        var pattern = square(2,n);
        //WriteLine(pattern);
        pattern = square(pattern,mod-2);
        
        WriteLine((anaaki * (long)pattern) %mod);
        
	
    }
    public static void swap(ref int a,ref int b){int temp = a;a= b;b = temp;}
    static void charswap(ref char a,ref char b){char temp = a;a= b;b = temp;}
    static int ncr(int n,int r){if(n<r)return 0;r = Min(r,n-r);long nn = 1;for(int i=n-r+1;i<=n;i++){nn = nn*i%mod;}long rr = 1;for(int i=1;i<=r;i++){rr = rr*i%mod;}rr = square((int)rr,mod-2);nn = nn * rr %mod;return (int)nn;}
    // a^b mod
    static int square(int a,int b){string binary = Convert.ToString(b,2);int bileng = binary.Length;long a_power = a;long value = 1;for(int i=bileng-1;i>=0;i--){if(binary[i] == '1'){value = value*a_power%mod;}a_power = a_power*a_power%mod;}return (int)value;}
    static int square2(int a,int b){long output = 1;var list = new List<long>();int sh = 1;long n = a;list.Add(a);while(sh < b){sh *= 2;n = n*n%mod;list.Add(n);}for(int i=list.Count-1;i>=0;i--){if(b > sh){b -= sh;sh /= 2;output = output*list[i]%mod;}}return (int)output;}
    //Šeí“Çæ
    static string rstr(){ return ReadLine(); }
    static int rint(){ return int.Parse(ReadLine()); }
    static long rlong(){ return long.Parse(ReadLine()); }
    static string[] stra(){ return ReadLine().Split(' '); }
    static char[] chara(){ string[] a=stra();string b="";for(int i=0;i<a.Length;i++){b+=a[i];}return b.ToCharArray();}
    static int[] inta(){ string[] read_str_array = ReadLine().Split(' '); int[] return_int_array = new int[read_str_array.Length]; for(int countup_i=0;countup_i<read_str_array.Length;countup_i++){ return_int_array[countup_i] = int.Parse(read_str_array[countup_i]); } return return_int_array; }
    static int[,] inta2(int num_array,int in_array){ int[,] int_array2 = new int[num_array,in_array]; for(int i=0;i<num_array;i++){ int[] temp_array = inta(); for(int j=0;j<in_array;j++){ int_array2[i,j] = temp_array[j]; } } return int_array2; }
    static long[] longa(){ string[] read_str_array = ReadLine().Split(' '); long[] return_long_array = new long[read_str_array.Length]; for(long countup_i=0;countup_i<read_str_array.Length;countup_i++){ return_long_array[countup_i] = long.Parse(read_str_array[countup_i]); } return return_long_array; }
    static double[] doublea(){ string[] read_str_array = ReadLine().Split(' '); double[] return_double_array = new double[read_str_array.Length]; for(long countup_i=0;countup_i<read_str_array.Length;countup_i++){ return_double_array[countup_i] = double.Parse(read_str_array[countup_i]); } return return_double_array; }
    // -----------------------------
    static long GCD(long a,long b){ if(a < b){ long temp = a; a = b; b = temp; } if(a % b == 0){ return b; } else{ long temp = b; b = a%b; a = temp; return GCD(a,b); } }
    static long LCM(long a,long b){ return a * b / GCD(a,b); }
    static void WriteArray(int[,] a,int b,int c){for(int i=0;i<b;i++){for(int j=0;j<c;j++){if(j!=0) Write(" ");Write(a[i,j]);}WriteLine();}}
}