Problem:
Score : 800 points

Problem Statement

There is a tree with N vertices. The vertices are numbered 1 through N. The i-th edge (1 \leq i \leq N - 1) connects Vertex x_i and y_i. For vertices v and w (1 \leq v, w \leq N), we will define the distance between v and w d(v, w) as "the number of edges contained in the path v-w".

A squirrel lives in each vertex of the tree. They are planning to move, as follows. First, they will freely choose a permutation of (1, 2, ..., N), p = (p_1, p_2, ..., p_N). Then, for each 1 \leq i \leq N, the squirrel that lived in Vertex i will move to Vertex p_i.

Since they like long travels, they have decided to maximize the total distance they traveled during the process. That is, they will choose p so that d(1, p_1) + d(2, p_2) + ... + d(N, p_N) will be maximized. How many such ways are there to choose p, modulo 10^9 + 7?

Constraints

2 \leq N \leq 5,000

1 \leq x_i, y_i \leq N

The input 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 choose p so that the condition is satisfied, modulo 10^9 + 7.

Sample Input 1

3
1 2
2 3

Sample Output 1

3

The maximum possible distance traveled by squirrels is 4.
There are three choices of p that achieve it, as follows:

(2, 3, 1)

(3, 1, 2)

(3, 2, 1)

Sample Input 2

4
1 2
1 3
1 4

Sample Output 2

11

The maximum possible distance traveled by squirrels is 6.
For example, p = (2, 1, 4, 3) achieves it.

Sample Input 3

6
1 2
1 3
1 4
2 5
2 6

Sample Output 3

36

Sample Input 4

7
1 2
6 3
4 5
1 7
1 5
2 3

Sample Output 4

396

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 Precalc ----------
#[allow(dead_code)]
struct Precalc {
    inv: Vec<ModInt>,
    fact: Vec<ModInt>,
    ifact: Vec<ModInt>,
}

#[allow(dead_code)]
impl Precalc {
    pub fn new(n: usize) -> Precalc {
        let mut inv = vec![ModInt::one(); n + 1];
        let mut fact = vec![ModInt::one(); n + 1];
        let mut ifact = vec![ModInt::one(); n + 1];
        for i in 2..(n + 1) {
            inv[i] = -inv[MOD as usize % i] * ModInt(MOD / i as u32);
            fact[i] = fact[i - 1] * ModInt(i as u32);
            ifact[i] = ifact[i - 1] * inv[i];
        }
        Precalc {
            inv: inv,
            fact: fact,
            ifact: ifact,
        }
    }
    pub fn inv(&self, n: usize) -> ModInt {
        self.inv[n]
    }
    pub fn fact(&self, n: usize) -> ModInt {
        self.fact[n]
    }
    pub fn ifact(&self, n: usize) -> ModInt {
        self.ifact[n]
    }
    pub fn comb(&self, n: usize, k: usize) -> ModInt {
        if k > n {
            return ModInt::zero();
        }
        self.fact[n] * self.ifact[k] * self.ifact[n - k]
    }
}
// ---------- end Precalc ----------
// ---------- 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 <= 16 {
        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 ----------

use std::cmp::*;
use std::io::Read;

fn calc(a: &[Vec<ModInt>]) -> Vec<ModInt> {
    if a.len() == 1 {
        a[0].clone()
    } else {
        let m = a.len() / 2;
        multiply(&calc(&a[..m]), &calc(&a[m..]), ModInt::zero())
    }
}

fn run() {
    let mut s = String::new();
    std::io::stdin().read_to_string(&mut s).unwrap();
    let mut it = s.trim().split_whitespace();
    let n: usize = it.next().unwrap().parse().unwrap();
    let mut g = vec![vec![]; n];
    for _ in 1..n {
        let x: usize = it.next().unwrap().parse().unwrap();
        let y: usize = it.next().unwrap().parse().unwrap();
        g[x - 1].push(y - 1);
        g[y - 1].push(x - 1);
    }
    let mut q = vec![];
    let mut parent = vec![0; n];
    let mut stack = vec![0];
    while let Some(v) = stack.pop() {
        q.push(v);
        for &u in g[v].iter() {
            if u != parent[v] {
                parent[u] = v;
                stack.push(u);
            }
        }
    }
    let mut size = vec![1; n];
    for &v in q.iter().rev() {
        for &u in g[v].iter() {
            if u != parent[v] {
                size[v] += size[u];
            }
        }
    }
    let pc = Precalc::new(n);
    let mut r = 0;
    loop {
        let mut c = (0, 0);
        for &u in g[r].iter() {
            if u != parent[r] {
                c = max(c, (size[u], u));
            }
        }
        if 2 * c.0 <= n {
            if 2 * c.0 == n {
                let ans = pc.fact(n / 2).pow(2);
                println!("{}", ans);
                return;
            }
            break;
        }
        r = c.1;
    }
    let mut a = vec![];
    for &u in g[r].iter() {
        if u == parent[r] {
            a.push(n - size[r]);
        } else {
            a.push(size[u]);
        }
    }
    let mut dp = vec![];
    for a in a {
        let mut mul = Vec::with_capacity(a + 1);
        for i in 0..(a + 1) {
            mul.push(pc.comb(a, i).pow(2) * pc.fact(i));
        }
        dp.push(mul);
    }
    let dp = calc(&dp);
    let mut ans = ModInt::zero();
    let mut sign = ModInt::one();
    for (i, v) in dp.iter().enumerate() {
        ans += sign * *v * pc.fact(n - i);
        sign = -sign;
    }
    println!("{}", ans);
}

fn main() {
    run();
}