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 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 <= 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 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 pc = Precalc::new(n);
    type Value = (usize, Vec<ModInt>);
    type Edge = ();
    let init = |_v: usize| -> Value { (1, vec![ModInt::one(), ModInt::one()]) };
    let merge = |a: Value, b: Value, _c: &Edge| -> Value {
        let u = min(b.0, n - b.0);
        let v = b.1[u];
        let mut next = vec![ModInt::zero(); a.0 + b.0 + 1];
        for (i, &a) in a.1.iter().enumerate() {
            let mut d = vec![ModInt::zero(); min(u, i) + 1];
            for j in 0..d.len() {
                d[j] = pc.comb(u, j) * pc.comb(i, j) * pc.fact(j);
            }
            let z = multiply(&d, &d, ModInt::zero());
            for k in 0..z.len() {
                next[i + u - k] += a * v * z[k];
            }
        }
        (a.0 + b.0, next)
    };
    let mut solver = TreeDP::new(n, init, merge);
    for _ in 1..n {
        let x: usize = it.next().unwrap().parse().unwrap();
        let y: usize = it.next().unwrap().parse().unwrap();
        solver.add_edge(x - 1, y - 1, ());
    }
    let ans = solver.solve(0).1[0];
    println!("{}", ans);
}

fn main() {
    run();
}