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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <cstdio>
#include <cstdlib>
#include <cassert>
#include <algorithm>
#include <functional>
#include <iostream>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <stack>
#include <string>
#include <vector>
#define repi(i,a,b) for(ll i=(a);i<(b);++i)
#define rep(i,a) repi(i,0,a)
#define repdi(i,a,b) for(ll i=(a)-1;i>=(b);--i)
#define repd(i,a) repdi(i,a,0)
#define itr(it,a) for( auto it = (a).begin(); it != (a).end(); ++it )
#define all(a) (a).begin(), (a).end()
#define rall(a) (a).rbegin(), (a).rend()

using ll = long long;
using P = std::pair<ll, ll>;

constexpr ll INF = 1ll<<60;

template<class T> inline bool chmax(T& a, T b) { if (a < b) { a = b; return 1; } return 0; }
template<class T> inline bool chmin(T& a, T b) { if (a > b) { a = b; return 1; } return 0; }

template<class S, class T>
std::ostream& operator<< ( std::ostream& out, const std::pair<S,T>& a )
{ std::cout << '(' << a.first << ", " << a.second << ')'; return out; }

template<class T>
std::ostream &operator<< ( std::ostream& out, const std::vector<T>& a )
{ std::cout << '['; rep( i, a.size() ){ std::cout << a[i]; if( i != a.size()-1 ) std::cout << ", "; } std::cout << ']'; return out; }

const ll mod = 1000000007;

struct mint {
  ll x; // typedef long long ll;
  mint(ll x=0):x((x%mod+mod)%mod){}
  mint operator-() const { return mint(-x);}
  mint& operator+=(const mint a) {
    if ((x += a.x) >= mod) x -= mod;
    return *this;
  }
  mint& operator-=(const mint a) {
    if ((x += mod-a.x) >= mod) x -= mod;
    return *this;
  }
  mint& operator*=(const mint a) {
    (x *= a.x) %= mod;
    return *this;
  }
  mint operator+(const mint a) const {
    mint res(*this);
    return res+=a;
  }
  mint operator-(const mint a) const {
    mint res(*this);
    return res-=a;
  }
  mint operator*(const mint a) const {
    mint res(*this);
    return res*=a;
  }
  mint pow(ll t) const {
    if (!t) return 1;
    mint a = pow(t>>1);
    a *= a;
    if (t&1) a *= *this;
    return a;
  }

  // for prime mod
  mint inv() const {
    return pow(mod-2);
  }
  mint& operator/=(const mint a) {
    return (*this) *= a.inv();
  }
  mint operator/(const mint a) const {
    mint res(*this);
    return res/=a;
  }
};

struct combination {
  std::vector<mint> fact, ifact;
  combination(int n):fact(n+1),ifact(n+1) {
    assert(n < mod);
    fact[0] = 1;
    for (int i = 1; i <= n; ++i) fact[i] = fact[i-1]*i;
    ifact[n] = fact[n].inv();
    for (int i = n; i >= 1; --i) ifact[i-1] = ifact[i]*i;
  }
  mint operator()(int n, int k) {
    if (k < 0 || k > n) return 0;
    return fact[n]*ifact[k]*ifact[n-k];
  }
};

ll N;
std::vector<ll> G[5010];
mint dp[5010][5010][2];
mint dp2[2][5010][2];
ll subtree[5010];
mint two[5010];
combination comb( 10000 );

void dfs( ll v, ll p = -1 )
{
  subtree[v] = 1;

  for( auto u : G[v] ) if( u != p )
  {
    dfs( u, v );
    subtree[v] += subtree[u];
  }

  ll sum = 0, cnt = 0;
  
  rep( j, subtree[v]+1 ) rep( k, 2 )
      dp2[0][j][k] = dp2[1][j][k] = 0;

  dp2[0][0][0] = 1;

  for( auto u : G[v] ) if( u != p )
  {
    rep( j, subtree[v]+1 ) rep( k, 2 )
      dp2[(cnt+1)&1][j][k] = 0;

    rep( sz, subtree[u] ) rep( j, sum+1 ) rep( k, 2 ) rep( l, 2 )
    {
      dp2[(cnt+1)&1][j+sz+1][k^l] += dp2[cnt&1][j][k]*dp[u][sz][l];

      if( !((sz+1)&1) )
        dp2[(cnt+1)&1][j][k^l^1] += dp2[cnt&1][j][k]*dp[u][sz][l]*comb.fact[sz]*two[sz/2]*comb.ifact[sz/2];
    }

    sum += subtree[u];
    ++cnt;
  }

  rep( j, sum+1 ) rep( k, 2 )
  {
    //printf( "dp2[%lld][%lld] = %lld\n", j, k, dp2[cnt&1][j][k].x );

    if( !v && !((j+1)&1) )
      dp[v][0][k] += dp2[cnt&1][j][k]*comb.fact[j+1]*two[(j+1)/2]*comb.ifact[(j+1)/2];
    else
      dp[v][j][k] += dp2[cnt&1][j][k];
  }
}

int main()
{
  std::cin >> N;

  rep( i, N-1 )
  {
    ll x, y;
    std::cin >> x >> y;
    --x; --y;

    G[x].emplace_back( y );
    G[y].emplace_back( x );
  }

  two[N+5] = mint(2).pow(N+5).inv();

  repd( i, N+5 )
    two[i] = two[i+1]*2;

  dfs( 0 );

  mint ans = 0;

  rep( j, subtree[0]+1 )
    ans += dp[0][j][0]-dp[0][j][1];

  std::cout << ans.x << std::endl;

  return 0;
}