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 Java8 (OpenJDK 1.8.0) (judge verdict: Time Limit Exceeded)


import java.util.Arrays;

public class Main {

  private static void solve() {
    int n = ni();
    int[] from = new int[n - 1];
    int[] to = new int[n - 1];
    for (int i = 0; i < n - 1; i++) {
      from[i] = ni() - 1;
      to[i] = ni() - 1;
    }
    int mod = (int) 1e9 + 7;
    long[] f2 = new long[n + 1];
    f2[1] = 1;
    for (int i = 3; i <= n; i += 2) {
      f2[i] = f2[i - 2] * i % mod;
    }

    int[][] g = packU(n, from, to);
    long[][][] dp = new long[n][2][n + 1];
    int[] num = new int[n];
    for (int i = 0; i < n; i++) {
      dp[i][0][1] = 1;
      num[i] = 1;
    }

    int[][] p3 = parents3(g, 0);
    int[] ord = p3[1];
    int[] par = p3[0];

    long[][] ndp = new long[2][n + 1];
    for (int k = n - 1; k > 0; k--) {
      int next = ord[k];
      int now = par[next];

      num[now] += num[next];
      for (int i = 1; i <= num[now]; i++) {
        ndp[0][i] = ndp[1][i] = 0;
        for (int j = 1; j <= num[next]; j++) {
          if (i > j) {
            ndp[0][i] += dp[next][0][j] * dp[now][0][i - j] % mod;
            ndp[0][i] += dp[next][1][j] * dp[now][1][i - j] % mod;
            ndp[1][i] += dp[next][0][j] * dp[now][1][i - j] % mod;
            ndp[1][i] += dp[next][1][j] * dp[now][0][i - j] % mod;
          }

          ndp[1][i] += dp[next][0][j] * dp[now][0][i] % mod * f2[j - 1] % mod;
          ndp[1][i] += dp[next][1][j] * dp[now][1][i] % mod * f2[j - 1] % mod;
          ndp[0][i] += dp[next][0][j] * dp[now][1][i] % mod * f2[j - 1] % mod;
          ndp[0][i] += dp[next][1][j] * dp[now][0][i] % mod * f2[j - 1] % mod;
          ndp[0][i] %= mod;
          ndp[1][i] %= mod;
        }
      }

      long[][] tmp = dp[now];
      dp[now] = ndp;
      ndp = tmp;
    }

    long ret = 0;
    for (int i = 1; i <= n; i++) {
      ret += (dp[0][0][i] + mod - dp[0][1][i]) * f2[i - 1];
      ret %= mod;
    }
    System.out.println(ret);
  }

  public static int[][] parents3(int[][] g, int root) {
    int n = g.length;
    int[] par = new int[n];
    Arrays.fill(par, -1);

    int[] depth = new int[n];
    depth[0] = 0;

    int[] q = new int[n];
    q[0] = root;
    for (int p = 0, r = 1; p < r; p++) {
      int cur = q[p];
      for (int nex : g[cur]) {
        if (par[cur] != nex) {
          q[r++] = nex;
          par[nex] = cur;
          depth[nex] = depth[cur] + 1;
        }
      }
    }
    return new int[][] {par, q, depth};
  }


  static int[][] packU(int n, int[] from, int[] to) {
    int[][] g = new int[n][];
    int[] p = new int[n];
    for (int f : from)
      p[f]++;
    for (int t : to)
      p[t]++;
    for (int i = 0; i < n; i++)
      g[i] = new int[p[i]];
    for (int i = 0; i < from.length; i++) {
      g[from[i]][--p[from[i]]] = to[i];
      g[to[i]][--p[to[i]]] = from[i];
    }
    return g;
  }

  public static void main(String[] args) {
    new Thread(null, new Runnable() {
      @Override
      public void run() {
        long start = System.currentTimeMillis();
        String debug = args.length > 0 ? args[0] : null;
        if (debug != null) {
          try {
            is = java.nio.file.Files.newInputStream(java.nio.file.Paths.get(debug));
          } catch (Exception e) {
            throw new RuntimeException(e);
          }
        }
        reader = new java.io.BufferedReader(new java.io.InputStreamReader(is), 32768);
        solve();
        out.flush();
        tr((System.currentTimeMillis() - start) + "ms");
      }
    }, "", 64000000).start();
  }

  private static java.io.InputStream is = System.in;
  private static java.io.PrintWriter out = new java.io.PrintWriter(System.out);
  private static java.util.StringTokenizer tokenizer = null;
  private static java.io.BufferedReader reader;

  public static String next() {
    while (tokenizer == null || !tokenizer.hasMoreTokens()) {
      try {
        tokenizer = new java.util.StringTokenizer(reader.readLine());
      } catch (Exception e) {
        throw new RuntimeException(e);
      }
    }
    return tokenizer.nextToken();
  }

  private static double nd() {
    return Double.parseDouble(next());
  }

  private static long nl() {
    return Long.parseLong(next());
  }

  private static int[] na(int n) {
    int[] a = new int[n];
    for (int i = 0; i < n; i++)
      a[i] = ni();
    return a;
  }

  private static char[] ns() {
    return next().toCharArray();
  }

  private static long[] nal(int n) {
    long[] a = new long[n];
    for (int i = 0; i < n; i++)
      a[i] = nl();
    return a;
  }

  private static int[][] ntable(int n, int m) {
    int[][] table = new int[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[i][j] = ni();
      }
    }
    return table;
  }

  private static int[][] nlist(int n, int m) {
    int[][] table = new int[m][n];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[j][i] = ni();
      }
    }
    return table;
  }

  private static int ni() {
    return Integer.parseInt(next());
  }

  private static void tr(Object... o) {
    if (is != System.in)
      System.out.println(java.util.Arrays.deepToString(o));
  }
}