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# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;

class F
{
	static void Main()
	{
		var n = int.Parse(Console.ReadLine());
		var rs = new int[n - 1].Select(_ => Console.ReadLine().Split().Select(int.Parse).ToArray()).ToArray();
		var map = rs.Concat(rs.Select(r => new[] { r[1], r[0] })).ToLookup(r => r[0], r => r[1]).ToDictionary(g => g.Key, g => g.ToArray());

		var tour = new int[n + 1].Select(_ => new List<int>()).ToArray();
		var c = 0;
		var q = new Stack<int>();
		var parents = new Stack<int>();
		var u = new bool[n + 1];

		q.Push(1);
		parents.Push(0);
		while (q.Any())
		{
			var p = q.Peek();
			if (!u[p])
			{
				u[p] = true;
				var p0 = parents.Peek();
				if (p != 1) tour[p0].Add(++c);
				foreach (var x in map[p])
				{
					if (x == p0) continue;
					q.Push(x);
				}
				parents.Push(p);
			}
			else
			{
				if (p != 1) tour[p].Add(++c);
				q.Pop();
				parents.Pop();
			}
		}

		// Time Test
	}
}