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: Accepted)

using System;
using System.Collections.Generic;
using CompLib.Mathematics;
using CompLib.Util;

public class Program
{
    private int N;
    private List<int>[] Edge;
    private List<int> Cnt;

    public void Solve()
    {
        var sc = new Scanner();
        N = sc.NextInt();
        Edge = new List<int>[N];
        for (int i = 0; i < N; i++)
        {
            Edge[i] = new List<int>();
        }

        for (int i = 0; i < N - 1; i++)
        {
            int s = sc.NextInt() - 1;
            int t = sc.NextInt() - 1;
            Edge[s].Add(t);
            Edge[t].Add(s);
        }

        // sの頂点の個数 - 黒の頂点の個数

        // 黒の頂点の個数の期待値 N/2
        // sの頂点の個数 = sの辺+1

        Cnt = new List<int>();
        int o;
        Calc(0, -1, out o);
        // 各辺がsに含まれるか?
        // 
        ModInt h = (ModInt.Mod + 1) / 2;
        ModInt sEdge = 0;
        foreach (var i in Cnt)
        {
            sEdge += (1 - ModInt.Pow(h, i)) * (1 - ModInt.Pow(h, N - i));
        }

        // 全部白 辺の個数0だけど頂点も0
        sEdge -= ModInt.Pow(h, N);

        Console.WriteLine(sEdge + 1 - N * h);
    }

    // 現在地、親、
    public void Calc(int n, int p, out int cnt)
    {
        cnt = 1;
        foreach (var to in Edge[n])
        {
            if (to == p) continue;
            int o;
            Calc(to, n, out o);
            Cnt.Add(o);
            cnt += o;
        }
    }

    public static void Main(string[] args) => new Program().Solve();
}

// https://bitbucket.org/camypaper/complib
namespace CompLib.Mathematics
{
    #region ModInt

    /// <summary>
    /// [0,<see cref="Mod"/>) までの値を取るような数
    /// </summary>
    public struct ModInt
    {
        /// <summary>
        /// 剰余を取る値．
        /// </summary>
        public const long Mod = (int) 1e9 + 7;

        /// <summary>
        /// 実際の数値．
        /// </summary>
        public long num;

        /// <summary>
        /// 値が <paramref name="n"/> であるようなインスタンスを構築します．
        /// </summary>
        /// <param name="n">インスタンスが持つ値</param>
        /// <remarks>パフォーマンスの問題上，コンストラクタ内では剰余を取りません．そのため，<paramref name="n"/> ∈ [0,<see cref="Mod"/>) を満たすような <paramref name="n"/> を渡してください．このコンストラクタは O(1) で実行されます．</remarks>
        public ModInt(long n)
        {
            num = n;
        }

        /// <summary>
        /// このインスタンスの数値を文字列に変換します．
        /// </summary>
        /// <returns>[0,<see cref="Mod"/>) の範囲内の整数を 10 進表記したもの．</returns>
        public override string ToString()
        {
            return num.ToString();
        }

        public static ModInt operator +(ModInt l, ModInt r)
        {
            l.num += r.num;
            if (l.num >= Mod) l.num -= Mod;
            return l;
        }

        public static ModInt operator -(ModInt l, ModInt r)
        {
            l.num -= r.num;
            if (l.num < 0) l.num += Mod;
            return l;
        }

        public static ModInt operator *(ModInt l, ModInt r)
        {
            return new ModInt(l.num * r.num % Mod);
        }

        public static implicit operator ModInt(long n)
        {
            n %= Mod;
            if (n < 0) n += Mod;
            return new ModInt(n);
        }

        /// <summary>
        /// 与えられた 2 つの数値からべき剰余を計算します．
        /// </summary>
        /// <param name="v">べき乗の底</param>
        /// <param name="k">べき指数</param>
        /// <returns>繰り返し二乗法により O(N log N) で実行されます．</returns>
        public static ModInt Pow(ModInt v, long k)
        {
            return Pow(v.num, k);
        }

        /// <summary>
        /// 与えられた 2 つの数値からべき剰余を計算します．
        /// </summary>
        /// <param name="v">べき乗の底</param>
        /// <param name="k">べき指数</param>
        /// <returns>繰り返し二乗法により O(N log N) で実行されます．</returns>
        public static ModInt Pow(long v, long k)
        {
            long ret = 1;
            for (k %= Mod - 1; k > 0; k >>= 1, v = v * v % Mod)
                if ((k & 1) == 1)
                    ret = ret * v % Mod;
            return new ModInt(ret);
        }

        /// <summary>
        /// 与えられた数の逆元を計算します．
        /// </summary>
        /// <param name="v">逆元を取る対象となる数</param>
        /// <returns>逆元となるような値</returns>
        /// <remarks>法が素数であることを仮定して，フェルマーの小定理に従って逆元を O(log N) で計算します．</remarks>
        public static ModInt Inverse(ModInt v)
        {
            return Pow(v, Mod - 2);
        }
    }

    #endregion

    #region Binomial Coefficient

    public class BinomialCoefficient
    {
        public ModInt[] fact, ifact;

        public BinomialCoefficient(int n)
        {
            fact = new ModInt[n + 1];
            ifact = new ModInt[n + 1];
            fact[0] = 1;
            for (int i = 1; i <= n; i++)
                fact[i] = fact[i - 1] * i;
            ifact[n] = ModInt.Inverse(fact[n]);
            for (int i = n - 1; i >= 0; i--)
                ifact[i] = ifact[i + 1] * (i + 1);
            ifact[0] = ifact[1];
        }

        public ModInt this[int n, int r]
        {
            get
            {
                if (n < 0 || n >= fact.Length || r < 0 || r > n) return 0;
                return fact[n] * ifact[n - r] * ifact[r];
            }
        }

        public ModInt RepeatedCombination(int n, int k)
        {
            if (k == 0) return 1;
            return this[n + k - 1, k];
        }
    }

    #endregion
}


namespace CompLib.Util
{
    using System;
    using System.Linq;

    class Scanner
    {
        private string[] _line;
        private int _index;
        private const char Separator = ' ';

        public Scanner()
        {
            _line = new string[0];
            _index = 0;
        }

        public string Next()
        {
            while (_index >= _line.Length)
            {
                _line = Console.ReadLine().Split(Separator);
                _index = 0;
            }

            return _line[_index++];
        }

        public int NextInt() => int.Parse(Next());
        public long NextLong() => long.Parse(Next());
        public double NextDouble() => double.Parse(Next());
        public decimal NextDecimal() => decimal.Parse(Next());
        public char NextChar() => Next()[0];
        public char[] NextCharArray() => Next().ToCharArray();

        public string[] Array()
        {
            _line = Console.ReadLine().Split(Separator);
            _index = _line.Length;
            return _line;
        }

        public int[] IntArray() => Array().Select(int.Parse).ToArray();
        public long[] LongArray() => Array().Select(long.Parse).ToArray();
        public double[] DoubleArray() => Array().Select(double.Parse).ToArray();
        public decimal[] DecimalArray() => Array().Select(decimal.Parse).ToArray();
    }
}