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)

#[doc = " https://github.com/hatoo/competitive-rust-snippets"]
#[allow(unused_imports)]
use std::cmp::{max, min, Ordering};
#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};
#[allow(unused_imports)]
use std::iter::FromIterator;
#[macro_export]
macro_rules ! chmax { ( $ x : expr , $ ( $ v : expr ) ,+ ) => { $ ( $ x = std :: cmp :: max ( $ x ,$ v ) ; ) + } ; }
#[macro_export]
macro_rules ! chmin { ( $ x : expr , $ ( $ v : expr ) ,+ ) => { $ ( $ x = std :: cmp :: min ( $ x ,$ v ) ; ) + } ; }
#[macro_export]
macro_rules ! max { ( $ x : expr ) => ( $ x ) ; ( $ x : expr , $ ( $ xs : expr ) ,+ ) => { std :: cmp :: max ( $ x , max ! ( $ ( $ xs ) ,+ ) ) } ; }
#[macro_export]
macro_rules ! min { ( $ x : expr ) => ( $ x ) ; ( $ x : expr , $ ( $ xs : expr ) ,+ ) => { std :: cmp :: min ( $ x , min ! ( $ ( $ xs ) ,+ ) ) } ; }
#[macro_export]
macro_rules ! dvec { ( $ t : expr ; $ len : expr ) => { vec ! [ $ t ; $ len ] } ; ( $ t : expr ; $ len : expr , $ ( $ rest : expr ) ,* ) => { vec ! [ dvec ! ( $ t ; $ ( $ rest ) ,* ) ; $ len ] } ; }
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { eprintln ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }
#[macro_export]
macro_rules ! input { ( source = $ s : expr , $ ( $ r : tt ) * ) => { let mut parser = Parser :: from_str ( $ s ) ; input_inner ! { parser , $ ( $ r ) * } } ; ( parser = $ parser : ident , $ ( $ r : tt ) * ) => { input_inner ! { $ parser , $ ( $ r ) * } } ; ( new_stdin_parser = $ parser : ident , $ ( $ r : tt ) * ) => { let stdin = std :: io :: stdin ( ) ; let reader = std :: io :: BufReader :: new ( stdin . lock ( ) ) ; let mut $ parser = Parser :: new ( reader ) ; input_inner ! { $ parser , $ ( $ r ) * } } ; ( $ ( $ r : tt ) * ) => { input ! { new_stdin_parser = parser , $ ( $ r ) * } } ; }
#[macro_export]
macro_rules ! input_inner { ( $ parser : ident ) => { } ; ( $ parser : ident , ) => { } ; ( $ parser : ident , $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let $ var = read_value ! ( $ parser , $ t ) ; input_inner ! { $ parser $ ( $ r ) * } } ; }
#[macro_export]
macro_rules ! read_value { ( $ parser : ident , ( $ ( $ t : tt ) ,* ) ) => { ( $ ( read_value ! ( $ parser , $ t ) ) ,* ) } ; ( $ parser : ident , [ $ t : tt ; $ len : expr ] ) => { ( 0 ..$ len ) . map ( | _ | read_value ! ( $ parser , $ t ) ) . collect ::< Vec < _ >> ( ) } ; ( $ parser : ident , chars ) => { read_value ! ( $ parser , String ) . chars ( ) . collect ::< Vec < char >> ( ) } ; ( $ parser : ident , usize1 ) => { read_value ! ( $ parser , usize ) - 1 } ; ( $ parser : ident , $ t : ty ) => { $ parser . next ::<$ t > ( ) . expect ( "Parse error" ) } ; }
use std::io;
use std::io::BufRead;
use std::str;
pub struct Parser<R> {
    reader: R,
    buf: Vec<u8>,
    pos: usize,
}
impl Parser<io::Empty> {
    pub fn from_str(s: &str) -> Parser<io::Empty> {
        Parser {
            reader: io::empty(),
            buf: s.as_bytes().to_vec(),
            pos: 0,
        }
    }
}
impl<R: BufRead> Parser<R> {
    pub fn new(reader: R) -> Parser<R> {
        Parser {
            reader: reader,
            buf: vec![],
            pos: 0,
        }
    }
    pub fn update_buf(&mut self) {
        self.buf.clear();
        self.pos = 0;
        loop {
            let (len, complete) = {
                let buf2 = self.reader.fill_buf().unwrap();
                self.buf.extend_from_slice(buf2);
                let len = buf2.len();
                if len == 0 {
                    break;
                }
                (len, buf2[len - 1] <= 0x20)
            };
            self.reader.consume(len);
            if complete {
                break;
            }
        }
    }
    pub fn next<T: str::FromStr>(&mut self) -> Result<T, T::Err> {
        loop {
            let mut begin = self.pos;
            while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                begin += 1;
            }
            let mut end = begin;
            while end < self.buf.len() && (self.buf[end] > 0x20) {
                end += 1;
            }
            if begin != self.buf.len() {
                self.pos = end;
                return str::from_utf8(&self.buf[begin..end]).unwrap().parse::<T>();
            } else {
                self.update_buf();
            }
        }
    }
}
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { eprintln ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }
#[doc = " https://github.com/hatoo/competitive-rust-snippets"]
const BIG_STACK_SIZE: bool = true;
#[allow(dead_code)]
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}
fn solve() {
    let out = stdout();
    let mut out = BufWriter::new(out.lock());
    input!{
        n:usize,
        ab:[(usize,usize);n-1],
    }
    let mut rr = ReRooting::<F>::new(n);
    for (a,b) in ab {
        rr.connect(a-1, b-1, 1, 1);
    }
    rr.build();
    let mut totok: Mod = 0.into();
    let mut two: Mod = 2.into();
    for u in 0..n {
        let mut adj = vec![];
        for i in 0..rr.g[u].len() {
            let e = rr.g[u][i];
            adj.push(e.dp as i64);
        }
        if adj.len() < 2 {
            continue;
        }
        let mut ng: Mod = 0.into();
        // 全白
        ng += 1;
        // 一つ黒含みで他全白
        for i in 0..adj.len() {
            // 全通りから全白を抜く
            ng += two.pow(adj[i]) - 1;
        }
        totok += two.pow((n-1) as i64) - ng;
    }
    // dbg!(totok);

    // 2^n x x = totok (mod M)
    let y = mod_inverse(two.pow(n as i64).0, 1_000_000_007);
    let x = totok * y;
    println!("{}", x);
}
#[doc = " ax=b(mod m)"]
#[doc = " ay=1 (mod m) -> y=a^{-1}"]
#[doc = " x=yb (mod m)"]
pub fn mod_inverse(a: i64, m: i64) -> i64 {
    let (_, x, _) = extgcd(a, m);
    (m + x % m) % m
}
#[doc = "ax+by=gcd(a,b) returns (gcd, x, y)"]
#[allow(dead_code)]
pub fn extgcd(a: i64, b: i64) -> (i64, i64, i64) {
    if b == 0 {
        (a, 1, 0)
    } else {
        let (gcd, x, y) = extgcd(b, a % b);
        (gcd, y, x - (a / b) * y)
    }
}
pub mod modular {
    const M: i64 = 1_000_000_007;
    #[derive(Debug, Clone, Copy, Default, PartialOrd, Ord, PartialEq, Eq)]
    pub struct Mod(pub i64);
    impl ::std::fmt::Display for Mod {
        fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
            write!(f, "{}", self.0)
        }
    }
    impl Mod {
        pub fn new(v: i64) -> Mod {
            Mod(v % M)
        }
        pub fn pow(self, mut r: i64) -> Mod {
            let mut k = self;
            let mut ret = 1.into();
            while r > 0 {
                if r % 2 != 0 {
                    ret = ret * k;
                }
                r /= 2;
                k = k * k;
            }
            ret
        }
        pub fn recip(self) -> Mod {
            self.pow(M - 2)
        }
    }
    use std::ops::*;
    impl<T: Into<Mod>> Add<T> for Mod {
        type Output = Mod;
        fn add(self, rhs: T) -> Self::Output {
            Mod::new(self.0 + rhs.into().0)
        }
    }
    impl<T: Into<Mod>> AddAssign<T> for Mod {
        fn add_assign(&mut self, rhs: T) {
            *self = *self + rhs;
        }
    }
    impl<T: Into<Mod>> Sub<T> for Mod {
        type Output = Mod;
        fn sub(self, rhs: T) -> Self::Output {
            Mod::new(self.0 - rhs.into().0 + M)
        }
    }
    impl<T: Into<Mod>> SubAssign<T> for Mod {
        fn sub_assign(&mut self, rhs: T) {
            *self = *self - rhs;
        }
    }
    impl<T: Into<Mod>> Mul<T> for Mod {
        type Output = Mod;
        fn mul(self, rhs: T) -> Self::Output {
            Mod::new(self.0 * rhs.into().0)
        }
    }
    impl<T: Into<Mod>> MulAssign<T> for Mod {
        fn mul_assign(&mut self, rhs: T) {
            *self = *self * rhs;
        }
    }
    impl<T: Into<Mod>> Div<T> for Mod {
        type Output = Mod;
        fn div(self, rhs: T) -> Self::Output {
            self * rhs.into().recip()
        }
    }
    impl<T: Into<Mod>> DivAssign<T> for Mod {
        fn div_assign(&mut self, rhs: T) {
            *self = *self / rhs;
        }
    }
    impl Neg for Mod {
        type Output = Mod;
        fn neg(self) -> Self::Output {
            Mod(0) - self
        }
    }
    impl<T: ::std::convert::Into<i64>> ::std::convert::From<T> for Mod {
        fn from(v: T) -> Self {
            Mod::new(v.into())
        }
    }
}
pub type Mod = modular::Mod;
struct F {}
impl Foldable for F {
    type Sum = usize;
    type T = usize;
    fn identity() -> usize {
        0
    }
    fn f(a: usize, b: usize) -> usize {
        a+b
    }
    fn g(x: usize, e: usize) -> usize {
        x+e
    }
}
trait Foldable {
    type Sum: Copy + std::fmt::Debug;
    type T: Copy + std::fmt::Debug;
    fn identity() -> Self::Sum;
    fn f(x: Self::Sum, y: Self::Sum) -> Self::Sum;
    fn g(acc: Self::Sum, x: Self::T) -> Self::Sum;
}
#[derive(Clone, Copy)]
struct Edge<T, Sum> {
    to: usize,
    data: T,
    dp: Sum,
    ndp: Sum,
}
struct ReRooting<F: Foldable> {
    subdp: Vec<F::Sum>,
    dp: Vec<F::Sum>,
    g: Vec<Vec<Edge<F::T, F::Sum>>>,
}
impl <F: Foldable> ReRooting<F> {
    pub fn new(n: usize) -> ReRooting<F> {
        ReRooting {
            subdp: vec![F::identity(); n],
            dp: vec![F::identity(); n],
            g: vec![vec![]; n],
        }
    }
    pub fn connect(&mut self, u: usize, v: usize, d: F::T, e: F::T) {
        self.g[u].push(Edge {
            to: v,
            data: d,
            dp: F::identity(),
            ndp: F::identity(),
        });
        self.g[v].push(Edge {
            to: u,
            data: e,
            dp: F::identity(),
            ndp: F::identity(),
        });
    }
    fn dfs_sub(&mut self, u: usize, par: Option<usize>) {
        for i in 0..self.g[u].len() {
            let e = self.g[u][i];
            if Some(e.to) == par { continue; }
            self.dfs_sub(e.to, Some(u));
            self.subdp[u] = F::f(self.subdp[u], F::g(self.subdp[e.to], e.data));
        }
    }
    fn dfs_all(&mut self, u: usize, par: Option<usize>, top: F::Sum) {
        let mut buf = F::identity();
        for i in 0..self.g[u].len() {
            let e = &mut self.g[u][i];
            // 左累積
            e.ndp = buf;
            e.dp = F::g(if Some(e.to) == par { top } else { self.subdp[e.to] }, e.data);
            buf = F::f(buf, e.dp);
        }
        self.dp[u] = buf;
        // 右累積
        buf = F::identity();
        for i in (0..self.g[u].len()).rev() {
            let e = self.g[u][i];
            if Some(e.to) != par {
                self.dfs_all(e.to, Some(u), F::f(e.ndp, buf));
            }
            let e = &mut self.g[u][i];
            e.ndp = F::f(e.ndp, buf);
            buf = F::f(buf, e.dp);
        }
    }
    pub fn build(&mut self) -> Vec<F::Sum> {
        self.dfs_sub(0, None);
        self.dfs_all(0, None, F::identity());
        self.dp.clone()
    }
}