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 Rust (1.15.1) (judge verdict: Accepted)

// ---------- begin ModInt ----------
const MOD: u32 = 1_000_000_007;

#[derive(Clone, Copy)]
struct ModInt(u32);

impl std::ops::Add for ModInt {
    type Output = ModInt;
    fn add(self, rhs: ModInt) -> Self::Output {
        let mut d = self.0 + rhs.0;
        if d >= MOD {
            d -= MOD;
        }
        ModInt(d)
    }
}

impl std::ops::AddAssign for ModInt {
    fn add_assign(&mut self, rhs: ModInt) {
        *self = *self + rhs;
    }
}

impl std::ops::Sub for ModInt {
    type Output = ModInt;
    fn sub(self, rhs: ModInt) -> Self::Output {
        let mut d = self.0 + MOD - rhs.0;
        if d >= MOD {
            d -= MOD;
        }
        ModInt(d)
    }
}

impl std::ops::SubAssign for ModInt {
    fn sub_assign(&mut self, rhs: ModInt) {
        *self = *self - rhs;
    }
}

impl std::ops::Mul for ModInt {
    type Output = ModInt;
    fn mul(self, rhs: ModInt) -> Self::Output {
        ModInt((self.0 as u64 * rhs.0 as u64 % MOD as u64) as u32)
    }
}

impl std::ops::MulAssign for ModInt {
    fn mul_assign(&mut self, rhs: ModInt) {
        *self = *self * rhs;
    }
}

impl std::ops::Neg for ModInt {
    type Output = ModInt;
    fn neg(self) -> Self::Output {
        ModInt(if self.0 == 0 {0} else {MOD - self.0})
    }
}

impl std::fmt::Display for ModInt {
    fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl std::str::FromStr for ModInt {
    type Err = std::num::ParseIntError;
    fn from_str(s: &str) -> Result<Self, Self::Err> {
        let val = s.parse::<u32>()?;
        Ok(ModInt::new(val))
    }
}

#[allow(dead_code)]
impl ModInt {
    pub fn new(n: u32) -> ModInt {
        ModInt(n % MOD)
    }
    pub fn zero() -> ModInt {
        ModInt(0)
    }
    pub fn one() -> ModInt {
        ModInt(1)
    }
    pub fn pow(self, mut n: u32) -> ModInt {
        let mut t = ModInt::one();
        let mut s = self;
        while n > 0 {
            if n & 1 == 1 {
                t *= s;
            }
            s *= s;
            n >>= 1;
        }
        t
    }
    pub fn inv(self) -> ModInt {
        self.pow(MOD - 2)
    }
}
// ---------- end ModInt ----------
// ---------- begin Tree DP ----------
struct TreeDP<Edge, Value, Init, Merge> {
    size: usize,
    graph: Vec<Vec<(usize, Edge)>>,
    init: Init,
    merge: Merge,
    phantom: std::marker::PhantomData<Value>,
}

impl<Edge, Value, Init, Merge> TreeDP<Edge, Value, Init, Merge>
where Edge: Clone,
      Value: Clone,
      Init: Fn(usize) -> Value,
      Merge: Fn(Value, Value, &Edge) -> Value,
{
    fn new(size: usize, init: Init, merge: Merge) -> Self {
        TreeDP {
            size: size,
            graph: vec![vec![]; size],
            init: init,
            merge: merge,
            phantom: std::marker::PhantomData,
        }
    }
    fn add_edge(&mut self, a: usize, b: usize, c: Edge) {
        assert!(a < self.size && b < self.size && a != b);
        self.graph[a].push((b, c.clone()));
        self.graph[b].push((a, c));
    }
    fn solve(&self, root: usize) -> Value {
        let size = self.size;
        let graph = &self.graph;
        let mut topo = vec![];
        let mut parent = vec![root; size];
        let mut stack = vec![root];
        while let Some(v) = stack.pop() {
            topo.push(v);
            for e in graph[v].iter() {
                if e.0 != parent[v] {
                    parent[e.0] = v;
                    stack.push(e.0);
                }
            }
        }
        assert!(topo.len() == size);
        let mut dp: Vec<Option<Value>> = (0..size).map(|_| None).collect();
        for &v in topo.iter().rev() {
            let mut now = (self.init)(v);
            for u in graph[v].iter() {
                if u.0 == parent[v] {
                    continue;
                }
                let b = dp[u.0].take().unwrap();
                now = (self.merge)(now, b, &u.1);
            }
            dp[v] = Some(now);
        }
        dp[root].take().unwrap()
    }
}
// ---------- end Tree DP ----------
// ---------- begin karatsuba multiplication ----------
fn karatsuba<T>(a: &[T], b: &[T], c: &mut [T], zero: T, buf: &mut [T]) 
where T: std::marker::Copy +
         std::ops::Add<Output = T> +
         std::ops::Sub<Output = T> +
         std::ops::Mul<Output = T> +
{
    let n = a.len();
    if n <= 32 {
        for (i, a) in a.iter().enumerate() {
            for (c, b) in c[i..].iter_mut().zip(b.iter()) {
                *c = *c + *a * *b;
            }
        }
        return;
    }
    if n & 1 == 1 {
        karatsuba(&a[1..], &b[1..], &mut c[2..], zero, buf);
        let x = a[0];
        let y = b[0];
        c[0] = c[0] + x * y;
        for (c, (a, b)) in c[1..].iter_mut().zip(a[1..].iter().zip(b[1..].iter())) {
            *c = *c + x * *b + *a * y;
        }
        return;
    }
    let m = n / 2;
    karatsuba(&a[..m], &b[..m], &mut c[..n], zero, buf);
    karatsuba(&a[m..], &b[m..], &mut c[n..], zero, buf);
    let (x, buf) = buf.split_at_mut(m);
    let (y, buf) = buf.split_at_mut(m);
    let (z, buf) = buf.split_at_mut(n);
    for z in z.iter_mut() {
        *z = zero;
    }
    for (x, (p, q)) in x.iter_mut().zip(a.iter().zip(a[m..].iter())) {
        *x = *p + *q;
    }
    for (y, (p, q)) in y.iter_mut().zip(b.iter().zip(b[m..].iter())) {
        *y = *p + *q;
    }
    karatsuba(x, y, z, zero, buf);
    for (z, (p, q)) in z.iter_mut().zip(c[..n].iter().zip(c[n..].iter())) {
        *z = *z - (*p + *q);
    }
    for (c, z) in c[m..].iter_mut().zip(z.iter()) {
        *c = *c + *z;
    }
}

fn multiply<T>(a: &[T], b: &[T], zero: T) -> Vec<T>
where T: std::marker::Copy +
         std::ops::Add<Output = T> +
         std::ops::Sub<Output = T> +
         std::ops::Mul<Output = T> +
{
    let mut i = 0;
    let mut j = 0;
    let mut ans = vec![zero; a.len() + b.len()];
    let mut buf = vec![zero; 4 * (a.len() + b.len())];
    let mut c = Vec::with_capacity(a.len() + b.len());
    while i < a.len() && j < b.len() {
        let x = a.len() - i;
        let y = b.len() - j;
        let z = std::cmp::min(x, y);
        c.clear();
        c.resize(2 * z, zero);
        karatsuba(&a[i..(i + z)], &b[j..(j + z)], &mut c, zero, &mut buf);
        for (ans, c) in ans[(i + j)..].iter_mut().zip(c.iter()) {
            *ans = *ans + *c;
        }
        if x <= y {
            j += x;
        } else {
            i += y;
        }
    }
    ans.truncate(a.len() + b.len() - 1);
    ans
}
// ---------- end karatsuba multiplication ----------
//https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8 より
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
    ($($r:tt)*) => {
        let s = {
            use std::io::Read;
            let mut s = String::new();
            std::io::stdin().read_to_string(&mut s).unwrap();
            s
        };
        let mut iter = s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
}

macro_rules! input_inner {
    ($iter:expr) => {};
    ($iter:expr, ) => {};
    ($iter:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($iter, $t);
        input_inner!{$iter $($r)*}
    };
}

macro_rules! read_value {
    ($iter:expr, ( $($t:tt),* )) => {
        ( $(read_value!($iter, $t)),* )
    };
    ($iter:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($iter, $t)).collect::<Vec<_>>()
    };
    ($iter:expr, chars) => {
        read_value!($iter, String).chars().collect::<Vec<char>>()
    };
    ($iter:expr, usize1) => {
        read_value!($iter, usize) - 1
    };
    ($iter:expr, $t:ty) => {
        $iter.next().unwrap().parse::<$t>().expect("Parse error")
    };
}

//

fn run() {
    input! {
        n: usize,
        e: [(usize1, usize1); n - 1],
    }
    let mut way = vec![ModInt::zero(); n + 1];
    way[2] = ModInt::one();
    for i in 4..(n + 1) {
        if i % 2 == 0 {
            way[i] = way[i - 2] * ModInt((i - 1) as u32);
        }
    }
    type Value = Vec<Vec<ModInt>>;
    type Edge = ();
    let init = |_v: usize| {
        let mut a = vec![vec![ModInt::zero(); 2]; 2];
        a[0][1] = ModInt::one();
        a
    };
    let merge = |a: Value, b: Value, _c: &Edge| -> Value {
        let mut next = vec![vec![ModInt::zero(); a[0].len() + b[0].len() - 1]; 2];
        for (x, a) in a.iter().enumerate() {
            for (y, b) in b.iter().enumerate() {
                let c = multiply(a, b, ModInt::zero());
                for (next, c) in next[x ^ y].iter_mut().zip(c.iter()) {
                    *next += *c;
                }
                let mut sum = ModInt::zero();
                for (j, b) in b.iter().enumerate() {
                    sum += *b * way[j];
                }
                for (next, a) in next[x ^ y ^ 1].iter_mut().zip(a.iter()) {
                    *next += *a * sum;
                }
            }
        }
        next
    };
    let mut solver = TreeDP::new(n, init, merge);
    for (a, b) in e {
        solver.add_edge(a, b, ());
    }
    let res = solver.solve(0);
    let mut ans = ModInt::zero();
    for (i, a) in res[0].iter().enumerate() {
        ans += *a * way[i];
    }
    for (i, a) in res[1].iter().enumerate() {
        ans -= *a * way[i];
    }
    println!("{}", ans);
}

fn main() {
    run();
}