Problem:
Score : 900 points

Problem Statement

Let N be an even number.

There is a tree with N vertices.
The vertices are numbered 1, 2, ..., N.
For each i (1 \leq i \leq N - 1), the i-th edge connects Vertex x_i and y_i.

Snuke would like to decorate the tree with ribbons, as follows.

First, he will divide the N vertices into N / 2 pairs.
Here, each vertex must belong to exactly one pair.
Then, for each pair (u, v), put a ribbon through all the edges contained in the shortest path between u and v.

Snuke is trying to divide the vertices into pairs so that the following condition is satisfied: "for every edge, there is at least one ribbon going through it."
How many ways are there to divide the vertices into pairs, satisfying this condition?
Find the count modulo 10^9 + 7.
Here, two ways to divide the vertices into pairs are considered different when there is a pair that is contained in one of the two ways but not in the other.

Constraints

N is an even number.

2 \leq N \leq 5000

1 \leq x_i, y_i \leq N

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
x_2 y_2
:
x_{N - 1} y_{N - 1}

Output

Print the number of the ways to divide the vertices into pairs, satisfying the condition, modulo 10^9 + 7.

Sample Input 1

4
1 2
2 3
3 4

Sample Output 1

2

There are three possible ways to divide the vertices into pairs, as shown below, and two satisfy the condition: the middle one and the right one.

Sample Input 2

4
1 2
1 3
1 4

Sample Output 2

3

There are three possible ways to divide the vertices into pairs, as shown below, and all of them satisfy the condition.

Sample Input 3

6
1 2
1 3
3 4
1 5
5 6

Sample Output 3

10

Sample Input 4

10
8 5
10 8
6 5
1 5
4 8
2 10
3 6
9 2
1 7

Sample Output 4

672

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

using System;
using System.Linq;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using SB = System.Text.StringBuilder;
using System.Numerics;
using static System.Math;
using Point = System.Numerics.Complex;
using Number = System.Int64;
namespace Program {
    public class Solver {
        Random rnd = new Random(0);
        public void Solve() {
            var n = ri;
            G = Enumerate(n, x => new List<int>());
            for (int i = 0; i < n - 1; i++)
            {
                var u = ri - 1;
                var v = ri - 1;
                G[u].Add(v);
                G[v].Add(u);
            }

            if (false)
            {
                n = 5000;
                G = Enumerate(n, x => new List<int>());
                for (int i = 1; i < n; i++)
                {
                    var p = rnd.Next() % i;
                    G[p].Add(i);
                    G[i].Add(p);
                }
            }
            coef = new ModInt[n + 1];
            coef[0] = 1;
            for (int i = 2; i <= n; i += 2)
                coef[i] = coef[i - 2] * (i - 1);
            dp = Enumerate(n, x => new long[0]);
            ep = Enumerate(n, x => new long[0]);
            dfs(-1, 0);
            ModInt ans = 0;
            for (int i = 0; i <= n; i += 2)
            {
                ans += dp[0][i] * coef[i];
                ans -= ep[0][i] * coef[i];
            }
            Console.WriteLine(ans);

        }
        ModInt[] coef;
        long[][] dp;
        long[][] ep;
        int dfs(int prev, int cur) {
            var ret = 1;
            var u = new long[2];
            var v = new long[2];
            u[1] = 1;
            foreach (var to in G[cur])
            {
                if (prev == to) continue;
                ret += dfs(cur, to);
                var nu = new long[ret + 1];
                var nv = new long[ret + 1];
                var p = dp[to];
                var q = ep[to];
                for (int i = 0; i < u.Length; i++)
                    for (int j = 0; j < dp[to].Length; j++)
                    {

                        nu[i + j] += u[i] * p[j];
                        nv[i + j] += u[i] * q[j];
                        nv[i + j] += v[i] * p[j];
                        nu[i + j] += v[i] * q[j];
                        if (j % 2 == 0)
                        {
                            nu[i] += (u[i] * q[j] + v[i] * p[j]) % ModInt.Mod * coef[j].num;
                            //nu[i] += (v[i] * p[j]) % ModInt.Mod * coef[j].num;
                            nv[i] += (u[i] * p[j] + v[i] * q[j]) % ModInt.Mod * coef[j].num;
                            //nv[i] += (v[i] * q[j]) % ModInt.Mod * coef[j].num;
                            nu[i] %= ModInt.Mod;
                            nv[i] %= ModInt.Mod;
                        }
                        nu[i + j] %= ModInt.Mod;
                        nv[i + j] %= ModInt.Mod;
                    }
                u = nu;
                v = nv;
            }
            dp[cur] = u;
            ep[cur] = v;
            return ret;
        }
        List<int>[] G;
        const long INF = 1L << 60;
        int ri { get { return sc.Integer(); } }
        long rl { get { return sc.Long(); } }
        double rd { get { return sc.Double(); } }
        string rs { get { return sc.Scan(); } }
        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());

        static T[] Enumerate<T>(int n, Func<int, T> f) {
            var a = new T[n];
            for (int i = 0; i < n; ++i) a[i] = f(i);
            return a;
        }
        static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
    }
}

#region main
static class Ex {
    static public string AsString(this IEnumerable<char> ie) { return new string(ie.ToArray()); }
    static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") {
        return string.Join(st, ie);
    }
    static public void Main() {
        Console.SetOut(new Program.IO.Printer(Console.OpenStandardOutput()) { AutoFlush = false });
        var solver = new Program.Solver();
        solver.Solve();
        Console.Out.Flush();
    }
}
#endregion
#region Ex
namespace Program.IO {
    using System.IO;
    using System.Text;
    using System.Globalization;

    public class Printer: StreamWriter {
        public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
        public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
    }

    public class StreamScanner {
        public StreamScanner(Stream stream) { str = stream; }

        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream { get { return isEof; } }

        private byte read() {
            if (isEof) return 0;
            if (ptr >= len)
            {
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) <= 0)
                {
                    isEof = true;
                    return 0;
                }
            }
            return buf[ptr++];
        }

        public char Char() {
            byte b = 0;
            do b = read(); while ((b < 33 || 126 < b) && !isEof);
            return (char)b;
        }
        public string Scan() {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read()) sb.Append(b);
            return sb.ToString();
        }
        public string ScanLine() {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n' && b != 0; b = (char)read()) if (b != '\r') sb.Append(b);
            return sb.ToString();
        }
        public long Long() { return isEof ? long.MinValue : long.Parse(Scan()); }
        public int Integer() { return isEof ? int.MinValue : int.Parse(Scan()); }
        public double Double() { return isEof ? double.NaN : double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }
}

#endregion

#region ModInt
public struct ModInt {
    public const long Mod = (long)1e9 + 7;
    public long num;
    public ModInt(long n) { num = n; }
    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); }
    public static ModInt Pow(ModInt v, long k) { return Pow(v.num, k); }
    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);
    }
    public static ModInt Inverse(ModInt v) { return Pow(v, Mod - 2); }
}
#endregion