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 Rust (1.15.1) (judge verdict: Accepted)

use std::cell::Cell;
use std::io::BufRead;

const MOD: i64 = 1000000007;

#[derive(Debug)]
struct Graph {
    n: usize,
    es: Vec<Vec<usize>>,
    p2: Vec<i64>,
    total: Cell<i64>,
}

impl Graph {
    fn new(n: usize) -> Graph {
        let mut p2 = vec![];
        let mut x = 1;
        for _ in 0..n + 1 {
            p2.push(x);
            x = x * 2 % MOD;
        }
        Graph {
            n: n,
            es: vec![vec![]; n],
            p2: p2,
            total: Cell::new(0),
        }
    }

    fn dfs(&self, u: usize, p: usize) -> usize {
        let mut chd = vec![];
        let mut tsz = 1;
        for v in &self.es[u] {
            if *v == p { continue; }
            let sz = self.dfs(*v, u);
            chd.push(sz);
            tsz += sz;
        }

        if u != p {
            chd.push(self.n - tsz);
        }

        if chd.len() > 1 {
            let mut sum = 1i64;
            for sz in &chd {
                sum += self.p2[*sz] - 1;
            }
            self.total.set((self.total.get() + (self.p2[self.n - 1] - sum)) % MOD);
        }
        tsz
    }
}

fn read_line(mut buf: &mut String) -> &str {
    buf.clear();
    std::io::stdin().read_line(&mut buf).unwrap();
    buf.trim_right()
}

fn main() {
    let mut buf = String::new();
    let n: usize = read_line(&mut buf).parse::<usize>().unwrap();

    let mut g = Graph::new(n);
    for _ in 0..n - 1 {
        let mut it = read_line(&mut buf).split_whitespace().map(|x| x.parse::<usize>().unwrap() - 1);
        let a = it.next().unwrap();
        let b = it.next().unwrap();
        g.es[a].push(b);
        g.es[b].push(a);
    }
    g.dfs(0, 0);

    let mut ans = g.total.get();
    for _ in 0..n {
        ans = ans * ((MOD + 1) / 2) % MOD;
    }
    ans = (ans + MOD) % MOD;
    println!("{}", ans);
}