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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include<iostream>
#include<vector>
using namespace std;
long mod=1e9+7;
long power(long a,long b){return b?power(a*a%mod,b/2)*(b%2?a:1)%mod:1;}
int N;
vector<int>G[2<<17];
long ans=0;
pair<long,long>dfs(int u,int p)
{
	vector<pair<long,long> >X;
	for(int v:G[u])
	{
		if(v!=p)
		{
			pair<long,long>p=dfs(v,u);
			X.push_back(p);
		}
	}
	if(X.empty())
	{
		return make_pair(0L,1L);
	}
	long sum=0;
	long prod=1;
	long w[3]={1,0,0};
	for(pair<long,long>p:X)
	{
		prod=prod*(p.second+1)%mod;
		w[2]=(w[2]*(p.second+1)+w[1]*p.second)%mod;
		w[1]=(w[1]+w[0]*p.second)%mod;
	}
	for(pair<long,long>p:X)
	{
		long now=prod*power(p.second+1,mod-2)%mod;
		sum+=now*p.first%mod+(now-1)*p.first%mod;
		sum=(sum%mod+mod)%mod;
	}
	sum+=w[2];sum%=mod;
	(ans+=sum)%=mod;
	pair<long,long>ret=make_pair(sum,(prod+prod-1+mod)%mod);
	for(pair<long,long>p:X)
	{
		(ret.first+=p.first+p.second)%=mod;
	}
	return ret;
}
main()
{
	cin>>N;
	for(int i=1;i<N;i++)
	{
		int u,v;cin>>u>>v;
		u--,v--;
		G[u].push_back(v);
		G[v].push_back(u);
	}
	dfs(0,-1);
	cout<<ans*power(power(2,N),mod-2)%mod<<endl;
}