Problem:
Max Score: $800$ Points

Problem Statement

There is an undirected connected graph with $N$ vertices and $N-1$ edges. The i-th edge connects u_i and v_i.

E869120 the coder moves in the graph as follows:

He move to adjacent vertex, but he can't a visit vertex two or more times.

He ends move when there is no way to move.

Otherwise, he moves randomly. (equal probability) If he has $p$ way to move this turn, he choose each vertex with $1/p$ probability.

Calculate the expected value of the number of turns, when E869120 starts from vertex i, for all i (1 ≤ i ≤ N).

Input

The input is given from standard input in the following format.

N
u_1 v_1
u_2 v_2
:
u_{N-1} v_{N-1}

Output

In i-th (1 ≤ i ≤ N) line, print the expected vaule of the number of turns E869120 moves.

The relative error or absolute error of output should be within 10^{-6}.

Constraints

$1 \le N \le 150,000$

The graph is connected.

Subtasks

Subtask 1 [ $190$ points ]

There is no vertex which degree is more than 2.

This means the graph looks like a list.

Subtask 2 [ $220$ points ]

1 ≤ N ≤ 1000.

Subtask 3 [ $390$ points ]

There are no additional constraints.

Sample Input 1

4
1 2
2 3
2 4

Sample Output 1

2.0
1.0
2.0
2.0

Sample Input 2

4
1 2
2 4
4 3

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Linq.Expressions;
using System.IO;
using System.Text;
using System.Diagnostics;

using Binary = System.Func<System.Linq.Expressions.ParameterExpression, System.Linq.Expressions.ParameterExpression, System.Linq.Expressions.BinaryExpression>;
using Unary = System.Func<System.Linq.Expressions.ParameterExpression, System.Linq.Expressions.UnaryExpression>;

class Program
{
    static StreamWriter sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
    static Scan sc = new Scan();
//    static Scan sc = new ScanCHK();
    const int M = 1000000007;
    const double eps = 1e-9;
    static readonly int[] dd = { 0, 1, 0, -1, 0 };
    static double[][] memo;
    static int[][] edge;
    static List<int>[] lis;
    static void Main()
    {
        int n = sc.Int;
        edge = new int[n - 1][];
        lis = new List<int>[n];
        memo = new double[n - 1][];
        for (int i = 0; i < n; i++)
        {
            lis[i] = new List<int>();
        }
        for (int i = 0; i < n - 1; i++)
        {
            memo[i] = new double[2];
            memo[i][0] = memo[i][1] = -1;
            edge[i] = sc.IntArr.Select(x => x - 1).ToArray();
            lis[edge[i][0]].Add(i);
            lis[edge[i][1]].Add(i);
        }
        var ans = new double[n];
        for (int i = n / 2; i < n; i++)
        {
            foreach (var item in lis[i])
            {
                ans[i] += dfs(item, edge[item][0] == i ? 1 : 0);
            }
            ans[i] /= lis[i].Count;
        }
        for (int i = n / 2 - 1; i >= 0; i--)
        {
            foreach (var item in lis[i])
            {
                ans[i] += dfs(item, edge[item][0] == i ? 1 : 0);
            }
            ans[i] /= lis[i].Count;
        }
        foreach (var item in ans)
        {
            Prt(item);
        }
        sw.Flush();
    }
    static double dfs(int j, int d)
    {
        if (memo[j][d] > -0.5) return memo[j][d];

        int nod = edge[j][d];
        if (lis[nod].Count == 1)
        {
            return memo[j][d] = 1;
        }
        double ret = 0;
        foreach (var item in lis[nod])
        {
            if (item == j) continue;

            ret += dfs(item, edge[item][0] == nod ? 1 : 0) + 1;
        }
        return memo[j][d] = ret / (lis[nod].Count - 1);
    }

    static void swap<T>(ref T a, ref T b) { var t = a; a = b; b = t; }
    static T Max<T>(params T[] a) { return a.Max(); }
    static T Min<T>(params T[] a) { return a.Min(); }
    static void DBG(string a) { Console.WriteLine(a); }
    static void DBG<T>(IEnumerable<T> a) { Console.WriteLine(string.Join(" ", a)); }
    static void DBG(params object[] a) { Console.WriteLine(string.Join(" ", a)); }
    static void Prt(string a) { sw.WriteLine(a); }
    static void Prt<T>(IEnumerable<T> a) { sw.WriteLine(string.Join(" ", a)); }
    static void Prt(params object[] a) { sw.WriteLine(string.Join(" ", a)); }
}
static class ex
{
    public static void swap<T>(this IList<T> a, int i, int j) { var t = a[i]; a[i] = a[j]; a[j] = t; }
    public static T[] copy<T>(this IList<T> a)
    {
        var ret = new T[a.Count];
        for (int i = 0; i < a.Count; i++) ret[i] = a[i];
        return ret;
    }
}
static class Operator<T>
{
    static readonly ParameterExpression x = Expression.Parameter(typeof(T), "x");
    static readonly ParameterExpression y = Expression.Parameter(typeof(T), "y");
    public static readonly Func<T, T, T> Add = Lambda(Expression.Add);
    public static readonly Func<T, T, T> Subtract = Lambda(Expression.Subtract);
    public static readonly Func<T, T, T> Multiply = Lambda(Expression.Multiply);
    public static readonly Func<T, T, T> Divide = Lambda(Expression.Divide);
    public static readonly Func<T, T> Plus = Lambda(Expression.UnaryPlus);
    public static readonly Func<T, T> Negate = Lambda(Expression.Negate);
    public static Func<T, T, T> Lambda(Binary op) { return Expression.Lambda<Func<T, T, T>>(op(x, y), x, y).Compile(); }
    public static Func<T, T> Lambda(Unary op) { return Expression.Lambda<Func<T, T>>(op(x), x).Compile(); }
}

class ScanCHK : Scan
{
    public new string Str { get { var s = Console.ReadLine(); if (s != s.Trim()) throw new Exception(); return s; } }
}
class Scan
{
    public int Int { get { return int.Parse(Str); } }
    public long Long { get { return long.Parse(Str); } }
    public double Double { get { return double.Parse(Str); } }
    public string Str { get { return Console.ReadLine().Trim(); } }
    public int[] IntArr { get { return StrArr.Select(int.Parse).ToArray(); } }
    public long[] LongArr { get { return StrArr.Select(long.Parse).ToArray(); } }
    public double[] DoubleArr { get { return StrArr.Select(double.Parse).ToArray(); } }
    public string[] StrArr { get { return Str.Split(); } }
    bool eq<T, U>() { return typeof(T).Equals(typeof(U)); }
    T ct<T, U>(U a) { return (T)Convert.ChangeType(a, typeof(T)); }
    T cv<T>(string s) { return eq<T, int>()    ? ct<T, int>(int.Parse(s))
                             : eq<T, long>()   ? ct<T, long>(long.Parse(s))
                             : eq<T, double>() ? ct<T, double>(double.Parse(s))
                             : eq<T, char>()   ? ct<T, char>(s[0])
                                               : ct<T, string>(s); }
    public void Multi<T>(out T a) { a = cv<T>(Str); }
    public void Multi<T, U>(out T a, out U b)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); }
    public void Multi<T, U, V>(out T a, out U b, out V c)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); }
    public void Multi<T, U, V, W>(out T a, out U b, out V c, out W d)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); }
    public void Multi<T, U, V, W, X>(out T a, out U b, out V c, out W d, out X e)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); e = cv<X>(ar[4]); }
}
class mymath
{
    public static long Mod = 1000000007;
    public static bool isprime(long a)
    {
        if (a < 2) return false;
        for (long i = 2; i * i <= a; i++) if (a % i == 0) return false;
        return true;
    }
    public static bool[] sieve(int n)
    {
        var p = new bool[n + 1];
        for (int i = 2; i <= n; i++) p[i] = true;
        for (int i = 2; i * i <= n; i++) if (p[i]) for (int j = i * i; j <= n; j += i) p[j] = false;
        return p;
    }
    public static List<int> getprimes(int n)
    {
        var prs = new List<int>();
        var p = sieve(n);
        for (int i = 2; i <= n; i++) if (p[i]) prs.Add(i);
        return prs;
    }
    public static long[][] E(int n)
    {
        var ret = new long[n][];
        for (int i = 0; i < n; i++) { ret[i] = new long[n]; ret[i][i] = 1; }
        return ret;
    }
    public static long[][] pow(long[][] A, long n)
    {
        if (n == 0) return E(A.Length);
        var t = pow(A, n / 2);
        if ((n & 1) == 0) return mul(t, t);
        return mul(mul(t, t), A);
    }
    public static double dot(double[] x, double[] y)
    {
        int n = x.Length;
        double ret = 0;
        for (int i = 0; i < n; i++) ret += x[i] * y[i];
        return ret;
    }
    public static long dot(long[] x, long[] y)
    {
        int n = x.Length;
        long ret = 0;
        for (int i = 0; i < n; i++) ret = (ret + x[i] * y[i]) % Mod;
        return ret;
    }
    public static T[][] trans<T>(T[][] A)
    {
        int n = A[0].Length, m = A.Length;
        var ret = new T[n][];
        for (int i = 0; i < n; i++) { ret[i] = new T[m]; for (int j = 0; j < m; j++) ret[i][j] = A[j][i]; }
        return ret;
    }
    public static double[] mul(double[][] A, double[] x)
    {
        int n = A.Length;
        var ret = new double[n];
        for (int i = 0; i < n; i++) ret[i] = dot(x, A[i]);
        return ret;
    }
    public static long[] mul(long[][] A, long[] x)
    {
        int n = A.Length;
        var ret = new long[n];
        for (int i = 0; i < n; i++) ret[i] = dot(x, A[i]);
        return ret;
    }
    public static long[][] mul(long[][] A, long[][] B)
    {
        int n = A.Length;
        var Bt = trans(B);
        var ret = new long[n][];
        for (int i = 0; i < n; i++) ret[i] = mul(Bt, A[i]);
        return ret;
    }
    public static long[] add(long[] x, long[] y)
    {
        int n = x.Length;
        var ret = new long[n];
        for (int i = 0; i < n; i++) ret[i] = (x[i] + y[i]) % Mod;
        return ret;
    }
    public static long[][] add(long[][] A, long[][] B)
    {
        int n = A.Length;
        var ret = new long[n][];
        for (int i = 0; i < n; i++) ret[i] = add(A[i], B[i]);
        return ret;
    }
    public static long pow(long a, long b)
    {
        if (a >= Mod) return pow(a % Mod, b);
        if (a == 0) return 0;
        if (b == 0) return 1;
        var t = pow(a, b / 2);
        if ((b & 1) == 0) return t * t % Mod;
        return t * t % Mod * a % Mod;
    }
    public static long inv(long a) { return pow(a, Mod - 2); }
    public static long gcd(long a, long b)
    {
        while (b > 0) { var t = a % b; a = b; b = t; }
        return a;
    }
    // a x + b y = gcd(a, b)
    public static long extgcd(long a, long b, out long x, out long y)
    {
        long g = a; x = 1; y = 0;
        if (b > 0) { g = extgcd(b, a % b, out y, out x); y -= a / b * x; }
        return g;
    }
    public static long lcm(long a, long b) { return a / gcd(a, b) * b; }
    public static long comb(int n, int r)
    {
        if (n < 0 || r < 0 || r > n) return 0;
        if (n - r < r) r = n - r;
        if (r == 0) return 1;
        if (r == 1) return n;
        int[] numer = new int[r], denom = new int[r];
        for (int k = 0; k < r; k++) { numer[k] = n - r + k + 1; denom[k] = k + 1; }
        for (int p = 2; p <= r; p++)
        {
            int piv = denom[p - 1];
            if (piv > 1)
            {
                int ofst = (n - r) % p;
                for (int k = p - 1; k < r; k += p) { numer[k - ofst] /= piv; denom[k] /= piv; }
            }
        }
        long ret = 1;
        for (int k = 0; k < r; k++) if (numer[k] > 1) ret = ret * numer[k] % Mod;
        return ret;
    }
}