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 System.IO;
using System.Linq;
using static System.Math;
using System.Text;
using System.Threading;
using System.Globalization;
using System.Runtime.CompilerServices;
using Library;

namespace Program
{
    public static class ABC149F
    {
        static public int numberOfRandomCases = 0;
        static public void MakeTestCase(List<string> _input, List<string> _output, ref Func<string[], bool> _outputChecker)
        {
        }
        static public void Solve()
        {
            var N = NN;
            var ABList = Repeat(0, N - 1).Select(_ => new { A = NN - 1, B = NN - 1 }).ToArray();
            var tree = new LIB_Tree(N);
            foreach (var item in ABList)
            {
                tree.AddPath(item.A, item.B);
            }
            var dp = tree.BuildTreeDP(idx => 1, (x, y) => x + y);
            LIB_Mod ans = 0;
            var pow = new LIB_Mod[N + 1];
            pow[0] = 1;
            for (var i = 1; i <= N; i++)
            {
                pow[i] = pow[i - 1] / 2;
            }
            for (var i = 0; i < N; i++)
            {
                LIB_Mod tmp = 1;
                LIB_Mod total = 1;
                foreach (var item in tree.GetSurround(i))
                {
                    tmp += (1 - pow[dp[item, i]]) / pow[dp[item, i]];
                    total *= pow[dp[item, i]];
                }
                ans += (1 - tmp * total) / 2;
            }
            Console.WriteLine(ans);
        }
        static class Console_
        {
            static Queue<string> param = new Queue<string>();
            public static string NextString() { if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item); return param.Dequeue(); }
        }
        class Printer : StreamWriter
        {
            public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
            public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { base.AutoFlush = false; }
            public Printer(Stream stream, Encoding encoding) : base(stream, encoding) { base.AutoFlush = false; }
        }
        static public void Main(string[] args) { if (args.Length == 0) { Console.SetOut(new Printer(Console.OpenStandardOutput())); } var t = new Thread(Solve, 134217728); t.Start(); t.Join(); Console.Out.Flush(); }
        static long NN => long.Parse(Console_.NextString());
        static double ND => double.Parse(Console_.NextString());
        static string NS => Console_.NextString();
        static long[] NNList(long N) => Repeat(0, N).Select(_ => NN).ToArray();
        static double[] NDList(long N) => Repeat(0, N).Select(_ => ND).ToArray();
        static string[] NSList(long N) => Repeat(0, N).Select(_ => NS).ToArray();
        static IEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => x.OrderBy(_ => xorshift);
        static long Count<T>(this IEnumerable<T> x, Func<T, bool> pred) => Enumerable.Count(x, pred);
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        static IEnumerable<int> Range(long s, long c) => Enumerable.Range((int)s, (int)c);
        static uint xorshift { get { _xsi.MoveNext(); return _xsi.Current; } }
        static IEnumerator<uint> _xsi = _xsc();
        static IEnumerator<uint> _xsc() { uint x = 123456789, y = 362436069, z = 521288629, w = (uint)(DateTime.Now.Ticks & 0xffffffff); while (true) { var t = x ^ (x << 11); x = y; y = z; z = w; w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); yield return w; } }
    }
}
namespace Library {
    struct LIB_Mod : IEquatable<LIB_Mod>, IEquatable<long>
    {
        static public long _mod = 1000000007; long v;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_Mod(long x)
        {
            if (x < _mod && x >= 0) v = x;
            else if ((v = x % _mod) < 0) v += _mod;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public implicit operator LIB_Mod(long x) => new LIB_Mod(x);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public implicit operator long(LIB_Mod x) => x.v;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Add(LIB_Mod x) { if ((v += x.v) >= _mod) v -= _mod; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Sub(LIB_Mod x) { if ((v -= x.v) < 0) v += _mod; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Mul(LIB_Mod x) => v = (v * x.v) % _mod;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Div(LIB_Mod x) => v = (v * Inverse(x.v)) % _mod;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Mod operator +(LIB_Mod x, LIB_Mod y) { var t = x.v + y.v; return t >= _mod ? new LIB_Mod { v = t - _mod } : new LIB_Mod { v = t }; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Mod operator -(LIB_Mod x, LIB_Mod y) { var t = x.v - y.v; return t < 0 ? new LIB_Mod { v = t + _mod } : new LIB_Mod { v = t }; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Mod operator *(LIB_Mod x, LIB_Mod y) => x.v * y.v;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Mod operator /(LIB_Mod x, LIB_Mod y) => x.v * Inverse(y.v);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool operator ==(LIB_Mod x, LIB_Mod y) => x.v == y.v;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool operator !=(LIB_Mod x, LIB_Mod y) => x.v != y.v;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long Inverse(long x)
        {
            long b = _mod, r = 1, u = 0, t = 0;
            while (b > 0)
            {
                var q = x / b;
                t = u;
                u = r - q * u;
                r = t;
                t = b;
                b = x - q * b;
                x = t;
            }
            return r < 0 ? r + _mod : r;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public bool Equals(LIB_Mod x) => v == x.v;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public bool Equals(long x) => v == x;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override bool Equals(object x) => x == null ? false : Equals((LIB_Mod)x);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override int GetHashCode() => v.GetHashCode();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override string ToString() => v.ToString();
        static List<LIB_Mod> _fact = new List<LIB_Mod>() { 1 };
        static long _factm = _mod;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static void B(long n)
        {
            if (_factm != _mod) _fact = new List<LIB_Mod>() { 1 };
            if (n >= _fact.Count)
                for (int i = _fact.Count; i <= n; ++i)
                    _fact.Add(_fact[i - 1] * i);
            _factm = _mod;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Mod Comb(long n, long k)
        {
            B(n);
            if (n == 0 && k == 0) return 1;
            if (n < k || n < 0) return 0;
            return _fact[(int)n] / _fact[(int)(n - k)] / _fact[(int)k];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Mod Perm(long n, long k)
        {
            B(n);
            if (n == 0 && k == 0) return 1;
            if (n < k || n < 0) return 0;
            return _fact[(int)n] / _fact[(int)(n - k)];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Mod Pow(LIB_Mod x, long y)
        {
            LIB_Mod a = 1;
            while (y != 0)
            {
                if ((y & 1) == 1) a.Mul(x);
                x.Mul(x);
                y >>= 1;
            }
            return a;
        }
    }
    class LIB_Tree
    {
        long N;
        List<long>[] path;
        public struct EulerRes
        {
            public long node;
            public long parent;
            public long direction;
        }
        List<EulerRes> eulerList;
        public struct BFSRes
        {
            public long node;
            public long parent;
        }
        List<BFSRes> bfsList;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_Tree(long n)
        {
            N = n;
            path = Enumerable.Repeat(0, (int)N).Select(_ => new List<long>()).ToArray();
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void AddPath(long u, long v)
        {
            if (u >= N || v >= N) throw new Exception();
            path[u].Add(v);
            path[v].Add(u);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public long[] GetSurround(long u) => path[u].ToArray();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public List<BFSRes> BFSFromRoot(long root)
        {
            bfsList = new List<BFSRes>();
            var q = new Queue<long>();
            var done = new bool[N];
            bfsList.Add(new BFSRes { node = root, parent = -1 });
            done[root] = true;
            q.Enqueue(root);
            while (q.Count > 0)
            {
                var w = q.Dequeue();
                foreach (var i in path[w])
                {
                    if (done[i]) continue;
                    done[i] = true;
                    q.Enqueue(i);
                    bfsList.Add(new BFSRes { node = i, parent = w });
                }
            }
            return bfsList;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public List<BFSRes> BFSFromLeaf(long root) => BFSFromRoot(root).ToArray().Reverse().ToList();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public List<EulerRes> EulerTour(long root)
        {
            eulerList = new List<EulerRes>();
            var s = new Stack<Tuple<long, long>>();
            var done = new bool[N];
            done[root] = true;
            s.Push(Tuple.Create(root, -1L));
            while (s.Count > 0)
            {
                var w = s.Peek();
                var ad = true;
                foreach (var i in path[w.Item1])
                {
                    if (done[i]) continue;
                    done[i] = true;
                    ad = false;
                    s.Push(Tuple.Create(i, w.Item1));
                }
                if (!ad || path[w.Item1].Count == 1) eulerList.Add(new EulerRes { node = w.Item1, parent = w.Item2, direction = 1 });
                if (ad)
                {
                    s.Pop();
                    eulerList.Add(new EulerRes { node = w.Item1, parent = w.Item2, direction = -1 });
                }
            }
            return eulerList;
        }
        public class LCAResult
        {
            int[] depth;
            int l;
            long[][] pr;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public LCAResult(int[] depth, int l, long[][] pr)
            {
                this.depth = depth;
                this.l = l;
                this.pr = pr;
            }
            public long this[long u, long v]
            {
                [MethodImpl(MethodImplOptions.AggressiveInlining)]
                get
                {
                    if (depth[u] > depth[v])
                    {
                        var t = u; u = v; v = t;
                    }
                    for (var k = 0; k < l; k++)
                    {
                        if ((((depth[v] - depth[u]) >> k) & 1) != 0) v = pr[k][v];
                    }
                    if (u == v) return u;
                    for (var k = l - 1; k >= 0; k--)
                    {
                        if (pr[k][u] != pr[k][v])
                        {
                            u = pr[k][u];
                            v = pr[k][v];
                        }
                    }
                    return pr[0][u];
                }
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LCAResult BuildLCA(long root)
        {
            var depth = new int[N];
            var l = 0;
            while (N > (1 << l)) l++;
            var pr = Enumerable.Repeat(0, l).Select(_ => new long[N]).ToArray();
            depth[root] = 0;
            pr[0][root] = -1;
            var q = new Stack<long>();
            q.Push(root);
            while (q.Count > 0)
            {
                var w = q.Pop();
                foreach (var i in path[w])
                {
                    if (i == pr[0][w]) continue;
                    q.Push(i);
                    depth[i] = depth[w] + 1;
                    pr[0][i] = w;
                }
            }
            for (var k = 0; k + 1 < l; k++)
            {
                for (var w = 0; w < N; w++)
                {
                    if (pr[k][w] < 0) pr[k + 1][w] = -1;
                    else pr[k + 1][w] = pr[k][pr[k][w]];
                }
            }
            return new LCAResult(depth, l, pr);
        }
        public class TreeDPResult<T>
        {
            Dictionary<long, T>[] dp;
            public TreeDPResult(Dictionary<long, T>[] dp)
            {
                this.dp = dp;
            }
            public T this[long vtx, long parent]
            {
                [MethodImpl(MethodImplOptions.AggressiveInlining)]
                get
                {
                    return dp[vtx][parent];
                }
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public TreeDPResult<T> BuildTreeDP<T>(Func<long, T> idxToVal, Func<T, T, T> f)
        {
            var dp = Enumerable.Repeat(0, (int)N).Select(_ => new Dictionary<long, T>()).ToArray();
            foreach (var item in BFSFromLeaf(0))
            {
                var acc = idxToVal(item.node);
                foreach (var item2 in path[item.node])
                {
                    if (item2 == item.parent) continue;
                    acc = f(acc, dp[item2][item.node]);
                }
                dp[item.node][item.parent] = acc;
            }
            var swag = new LIB_SlidingWindowAggregation<T>(f);
            var done = new bool[N];
            var q = new Queue<long>();
            done[0] = true;
            q.Enqueue(0);
            foreach (var item in path[0]) swag.PushBack(dp[item][0]);
            swag.PushBack(idxToVal(0));
            foreach (var item in path[0])
            {
                swag.PopFront();
                dp[0][item] = swag.Aggregate();
                swag.PushBack(dp[item][0]);
            }
            while (q.Count > 0)
            {
                var v = q.Dequeue();
                foreach (var item in path[v])
                {
                    if (done[item]) continue;
                    done[item] = true;
                    q.Enqueue(item);
                    var val = idxToVal(item);
                    if (path[item].Count == 1)
                    {
                        var x = path[item][0];
                        if (x != v) dp[item][x] = val;
                    }
                    else if (path[item].Count == 2)
                    {
                        var x = path[item][0]; var y = path[item][1];
                        var xval = f(val, dp[x][item]); var yval = f(val, dp[y][item]);
                        if (x != v) dp[item][x] = yval;
                        if (y != v) dp[item][y] = xval;
                    }
                    else if (path[item].Count == 3)
                    {
                        var x = path[item][0]; var y = path[item][1]; var z = path[item][2];
                        var yzval = f(val, f(dp[y][item], dp[z][item])); var xzval = f(val, f(dp[x][item], dp[z][item])); var xyval = f(val, f(dp[x][item], dp[y][item]));
                        if (x != v) dp[item][x] = yzval;
                        if (y != v) dp[item][y] = xzval;
                        if (z != v) dp[item][z] = xyval;
                    }
                    else if (path[item].Count == 4)
                    {
                        var x = path[item][0]; var y = path[item][1]; var z = path[item][2]; var w = path[item][3];
                        var xyval = f(dp[x][item], dp[y][item]); var zwval = f(dp[z][item], dp[w][item]);
                        var yzwval = f(val, f(dp[y][item], zwval)); var xzwval = f(val, f(dp[x][item], zwval)); var xywval = f(val, f(xyval, dp[w][item])); var xyzval = f(val, f(xyval, dp[z][item]));
                        if (x != v) dp[item][x] = yzwval;
                        if (y != v) dp[item][y] = xzwval;
                        if (z != v) dp[item][z] = xywval;
                        if (w != v) dp[item][w] = xyzval;
                    }
                    else
                    {
                        swag.Clear();
                        foreach (var item2 in path[item]) swag.PushBack(dp[item2][item]);
                        swag.PushBack(val);
                        foreach (var item2 in path[item])
                        {
                            swag.PopFront();
                            if (item2 != v) dp[item][item2] = swag.Aggregate();
                            swag.PushBack(dp[item2][item]);
                        }
                    }
                }
            }
            return new TreeDPResult<T>(dp);
        }
    }
    class LIB_SlidingWindowAggregation<T>
    {
        T[][] list;
        T[][] agg;
        int[] size;
        Func<T, T, T> fun;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_SlidingWindowAggregation(IEnumerable<T> ary, Func<T, T, T> f)
        {
            list = new T[2][]; agg = new T[2][]; size = new int[2];
            fun = f;
            list[0] = new T[16]; agg[0] = new T[16]; size[0] = 0;
            if (ary.Any())
            {
                var temp = ary.ToArray();
                size[1] = temp.Length;
                list[1] = new T[size[1]];
                agg[1] = new T[size[1]];
                for (var i = 0; i < list[1].Length; i++)
                {
                    list[1][i] = temp[i];
                    if (i == 0) agg[1][i] = temp[i];
                    else agg[1][i] = fun(agg[1][i - 1], temp[i]);
                }
            }
            else
            {
                list[1] = new T[16]; agg[1] = new T[16]; size[1] = 0;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_SlidingWindowAggregation(Func<T, T, T> f) : this(new T[0], f) { }
        public int Count => size[0] + size[1];
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Clear()
        {
            size[0] = size[1] = 0;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        void Push(int lista, T val)
        {
            if (list[lista].Length == size[lista])
            {
                var newAry = new T[size[lista] * 2];
                var newAgg = new T[size[lista] * 2];
                Array.Copy(list[lista], newAry, size[lista]);
                Array.Copy(agg[lista], newAgg, size[lista]);
                list[lista] = newAry;
                agg[lista] = newAgg;
            }
            if (size[lista] == 0) agg[lista][0] = val;
            else if (lista == 0) agg[lista][size[lista]] = fun(val, agg[lista][size[lista] - 1]);
            else agg[lista][size[lista]] = fun(agg[lista][size[lista] - 1], val);
            list[lista][size[lista]++] = val;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void PushBack(T val) => Push(1, val);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void PushFront(T val) => Push(0, val);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T Pop(int lista)
        {
            var listb = 1 - lista;
            if (size[lista] == 0)
            {
                if (size[listb] == 0) throw new Exception();
                var mid = (size[listb] - 1) / 2;
                if (list[lista].Length <= mid)
                {
                    list[lista] = new T[mid + 1];
                    agg[lista] = new T[mid + 1];
                }
                size[lista] = 0;
                for (var i = mid; i >= 0; i--)
                {
                    if (size[lista] == 0) agg[lista][size[lista]] = list[listb][i];
                    else if (lista == 0) agg[lista][size[lista]] = fun(list[listb][i], agg[lista][size[lista] - 1]);
                    else agg[lista][size[lista]] = fun(agg[lista][size[lista] - 1], list[listb][i]);
                    list[lista][size[lista]++] = list[listb][i];
                }
                for (var i = mid + 1; i < size[listb]; i++)
                {
                    var idx = i - mid - 1;
                    if (idx == 0) agg[listb][idx] = list[listb][i];
                    else if (lista == 0) agg[listb][idx] = fun(list[listb][i], agg[listb][idx - 1]);
                    else agg[listb][idx] = fun(agg[listb][idx - 1], list[listb][i]);
                    list[listb][idx] = list[listb][i];
                }
                size[listb] -= size[lista];
            }
            return list[lista][--size[lista]];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T PopBack() => Pop(1);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T PopFront() => Pop(0);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T Aggregate()
        {
            if (size[0] == 0 && size[1] == 0) throw new Exception();
            else if (size[1] == 0) return agg[0][size[0] - 1];
            else if (size[0] == 0) return agg[1][size[1] - 1];
            else return fun(agg[0][size[0] - 1], agg[1][size[1] - 1]);
        }
    }
}