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 (1.42.0) (judge verdict: Accepted)

struct DFS {
    n: usize,
    g: Vec<Vec<usize>>,
    dp: Vec<Vec<usize>>,
}

impl DFS {
    fn new(n: usize) -> DFS {
        DFS {
            n,
            g: vec![Vec::new(); n],
            dp: vec![Vec::new(); n],
        }
    }

    fn add_edge(&mut self, from: usize, to: usize) {
        self.g[from].push(to);
        self.dp[from].push(to);
    }

    fn search(&mut self, par: usize, from: usize) -> usize {
        let mut sum = 0;
        let mut par_i = !0;
        for (i, &to) in self.g[from].clone().iter().enumerate() {
            if to == par {
                par_i = i;
                continue;
            }
            self.dp[from][i] = self.search(from, to);
            sum += self.dp[from][i];
        }
        if par_i != !0 {
            self.dp[from][par_i] = self.n - (sum + 1);
        }
        sum + 1
    }
}

fn main() {
    const MOD: usize = 1_000_000_007;

    let n: usize = {
        let mut buf = String::new();
        std::io::stdin().read_line(&mut buf).unwrap();
        buf.trim_end().parse().unwrap()
    };

    let mut dfs = DFS::new(n);
    for _ in 0..(n - 1) {
        let (a, b) = {
            let mut buf = String::new();
            std::io::stdin().read_line(&mut buf).unwrap();
            let mut iter = buf.split_whitespace();
            (
                iter.next().unwrap().parse::<usize>().unwrap() - 1,
                iter.next().unwrap().parse::<usize>().unwrap() - 1,
            )
        };

        dfs.add_edge(a, b);
        dfs.add_edge(b, a);
    }
    dfs.search(!0, 0);

    let pow = |mut a, mut n| {
        let mut res = 1;
        while n > 0 {
            if n & 1 > 0 {
                res = res * a % MOD;
            }
            a = a * a % MOD;
            n >>= 1;
        }
        res
    };

    let mut ans = 0;
    for i in 0..n {
        let mut res = 1;
        res = (res + MOD - pow(pow(2, n - 1), MOD - 2)) % MOD;
        let mut sum = 0;
        for &x in dfs.dp[i].iter() {
            let p = pow(pow(2, n - 1 - x), MOD - 2);
            let q = (1 + MOD - pow(pow(2, x), MOD - 2)) % MOD;
            sum = (sum + (p * q % MOD)) % MOD;
        }
        res = (res + MOD - sum) % MOD;
        ans = (ans + res * pow(2, MOD - 2)) % MOD;
    }
    println!("{}", ans);
}