#[allow(unused_imports)] use std::cmp::*; // https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8 macro_rules! input { ($($r:tt)*) => { let stdin = std::io::stdin(); let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock())); let mut next = move || -> String{ bytes.by_ref().map(|r|r.unwrap() as char) .skip_while(|c|c.is_whitespace()) .take_while(|c|!c.is_whitespace()) .collect() }; input_inner!{next, $($r)*} }; } macro_rules! input_inner { ($next:expr) => {}; ($next:expr,) => {}; ($next:expr, $var:ident : $t:tt $($r:tt)*) => { let $var = read_value!($next, $t); input_inner!{$next $($r)*} }; } macro_rules! read_value { ($next:expr, ( $($t:tt),* )) => { ($(read_value!($next, $t)),*) }; ($next:expr, [ $t:tt ; $len:expr ]) => { (0..$len).map(|_| read_value!($next, $t)).collect::>() }; ($next:expr, chars) => { read_value!($next, String).chars().collect::>() }; ($next:expr, usize1) => (read_value!($next, usize) - 1); ($next:expr, [ $t:tt ]) => {{ let len = read_value!($next, usize); read_value!($next, [$t; len]) }}; ($next:expr, $t:ty) => ($next().parse::<$t>().expect("Parse error")); } trait Change { fn chmax(&mut self, x: Self); fn chmin(&mut self, x: Self); } impl Change for T { fn chmax(&mut self, x: T) { if *self < x { *self = x; } } fn chmin(&mut self, x: T) { if *self > x { *self = x; } } } /// Verified by https://atcoder.jp/contests/abc198/submissions/21774342 mod mod_int { use std::ops::*; pub trait Mod: Copy { fn m() -> i64; } #[derive(Copy, Clone, Hash, PartialEq, Eq, PartialOrd, Ord)] pub struct ModInt { pub x: i64, phantom: ::std::marker::PhantomData } impl ModInt { // x >= 0 pub fn new(x: i64) -> Self { ModInt::new_internal(x % M::m()) } fn new_internal(x: i64) -> Self { ModInt { x: x, phantom: ::std::marker::PhantomData } } pub fn pow(self, mut e: i64) -> Self { debug_assert!(e >= 0); let mut sum = ModInt::new_internal(1); let mut cur = self; while e > 0 { if e % 2 != 0 { sum *= cur; } cur *= cur; e /= 2; } sum } #[allow(dead_code)] pub fn inv(self) -> Self { self.pow(M::m() - 2) } } impl>> Add for ModInt { type Output = Self; fn add(self, other: T) -> Self { let other = other.into(); let mut sum = self.x + other.x; if sum >= M::m() { sum -= M::m(); } ModInt::new_internal(sum) } } impl>> Sub for ModInt { type Output = Self; fn sub(self, other: T) -> Self { let other = other.into(); let mut sum = self.x - other.x; if sum < 0 { sum += M::m(); } ModInt::new_internal(sum) } } impl>> Mul for ModInt { type Output = Self; fn mul(self, other: T) -> Self { ModInt::new(self.x * other.into().x % M::m()) } } impl>> AddAssign for ModInt { fn add_assign(&mut self, other: T) { *self = *self + other; } } impl>> SubAssign for ModInt { fn sub_assign(&mut self, other: T) { *self = *self - other; } } impl>> MulAssign for ModInt { fn mul_assign(&mut self, other: T) { *self = *self * other; } } impl Neg for ModInt { type Output = Self; fn neg(self) -> Self { ModInt::new(0) - self } } impl ::std::fmt::Display for ModInt { fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { self.x.fmt(f) } } impl ::std::fmt::Debug for ModInt { fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { let (mut a, mut b, _) = red(self.x, M::m()); if b < 0 { a = -a; b = -b; } write!(f, "{}/{}", a, b) } } impl From for ModInt { fn from(x: i64) -> Self { Self::new(x) } } // Finds the simplest fraction x/y congruent to r mod p. // The return value (x, y, z) satisfies x = y * r + z * p. fn red(r: i64, p: i64) -> (i64, i64, i64) { if r.abs() <= 10000 { return (r, 1, 0); } let mut nxt_r = p % r; let mut q = p / r; if 2 * nxt_r >= r { nxt_r -= r; q += 1; } if 2 * nxt_r <= -r { nxt_r += r; q -= 1; } let (x, z, y) = red(nxt_r, r); (x, y - q * z, z) } } // mod mod_int macro_rules! define_mod { ($struct_name: ident, $modulo: expr) => { #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)] struct $struct_name {} impl mod_int::Mod for $struct_name { fn m() -> i64 { $modulo } } } } const MOD: i64 = 998_244_353; define_mod!(P, MOD); type MInt = mod_int::ModInt

; // Depends on MInt.rs fn fact_init(w: usize) -> (Vec, Vec) { let mut fac = vec![MInt::new(1); w]; let mut invfac = vec![0.into(); w]; for i in 1 .. w { fac[i] = fac[i - 1] * i as i64; } invfac[w - 1] = fac[w - 1].inv(); for i in (0 .. w - 1).rev() { invfac[i] = invfac[i + 1] * (i as i64 + 1); } (fac, invfac) } // FFT (in-place, verified as NTT only) // R: Ring + Copy // Verified by: https://judge.yosupo.jp/submission/53831 // Adopts the technique used in https://judge.yosupo.jp/submission/3153. mod fft { use std::ops::*; // n should be a power of 2. zeta is a primitive n-th root of unity. // one is unity // Note that the result is bit-reversed. pub fn fft(f: &mut [R], zeta: R, one: R) where R: Copy + Add + Sub + Mul { let n = f.len(); assert!(n.is_power_of_two()); let mut m = n; let mut base = zeta; unsafe { while m > 2 { m >>= 1; let mut r = 0; while r < n { let mut w = one; for s in r..r + m { let &u = f.get_unchecked(s); let d = *f.get_unchecked(s + m); *f.get_unchecked_mut(s) = u + d; *f.get_unchecked_mut(s + m) = w * (u - d); w = w * base; } r += 2 * m; } base = base * base; } if m > 1 { // m = 1 let mut r = 0; while r < n { let &u = f.get_unchecked(r); let d = *f.get_unchecked(r + 1); *f.get_unchecked_mut(r) = u + d; *f.get_unchecked_mut(r + 1) = u - d; r += 2; } } } } pub fn inv_fft(f: &mut [R], zeta_inv: R, one: R) where R: Copy + Add + Sub + Mul { let n = f.len(); assert!(n.is_power_of_two()); let zeta = zeta_inv; // inverse FFT let mut zetapow = Vec::with_capacity(20); { let mut m = 1; let mut cur = zeta; while m < n { zetapow.push(cur); cur = cur * cur; m *= 2; } } let mut m = 1; unsafe { if m < n { zetapow.pop(); let mut r = 0; while r < n { let &u = f.get_unchecked(r); let d = *f.get_unchecked(r + 1); *f.get_unchecked_mut(r) = u + d; *f.get_unchecked_mut(r + 1) = u - d; r += 2; } m = 2; } while m < n { let base = zetapow.pop().unwrap(); let mut r = 0; while r < n { let mut w = one; for s in r..r + m { let &u = f.get_unchecked(s); let d = *f.get_unchecked(s + m) * w; *f.get_unchecked_mut(s) = u + d; *f.get_unchecked_mut(s + m) = u - d; w = w * base; } r += 2 * m; } m *= 2; } } } } /// Computes f^{-1} mod x^{f.len()}. /// /// Reference: https://codeforces.com/blog/entry/56422 /// /// Complexity: O(n log n) /// /// Verified by: https://judge.yosupo.jp/submission/3219 /// /// Depends on: MInt.rs, fft.rs fn formal_power_series_inv( f: &[mod_int::ModInt

], gen: mod_int::ModInt

, ) -> Vec> { let n = f.len(); assert!(n.is_power_of_two()); assert_eq!(f[0], 1.into()); let mut sz = 1; let mut r = vec![mod_int::ModInt::new(0); n]; let mut tmp_f = vec![mod_int::ModInt::new(0); n]; let mut tmp_r = vec![mod_int::ModInt::new(0); n]; r[0] = 1.into(); // Adopts the technique used in https://judge.yosupo.jp/submission/3153 while sz < n { let zeta = gen.pow((P::m() - 1) / sz as i64 / 2); for i in 0..2 * sz { tmp_f[i] = f[i]; tmp_r[i] = r[i]; } fft::fft(&mut tmp_r[..2 * sz], zeta, 1.into()); fft::fft(&mut tmp_f[..2 * sz], zeta, 1.into()); let fac = mod_int::ModInt::new(2 * sz as i64).inv().pow(2); for i in 0..2 * sz { tmp_f[i] = tmp_f[i] * tmp_r[i]; } fft::inv_fft(&mut tmp_f[..2 * sz], zeta.inv(), 1.into()); for i in 0..sz { tmp_f[i] = 0.into(); } fft::fft(&mut tmp_f[..2 * sz], zeta, 1.into()); for i in 0..2 * sz { tmp_f[i] = -tmp_f[i] * tmp_r[i] * fac; } fft::inv_fft(&mut tmp_f[..2 * sz], zeta.inv(), 1.into()); for i in sz..2 * sz { r[i] = tmp_f[i]; } sz *= 2; } r } /// Computes ln f mod x^{f.len()}. /// /// Reference: https://codeforces.com/blog/entry/56422 /// /// Complexity: O(n log n) /// /// Verified by: https://judge.yosupo.jp/submission/53708 /// /// Depends on: MInt.rs, fact_init.rs, fft.rs, formal_power_series_inv fn formal_power_series_log( f: &[mod_int::ModInt

], gen: mod_int::ModInt

, fac: &[mod_int::ModInt

], invfac: &[mod_int::ModInt

], ) -> Vec> { let n = f.len(); assert!(n.is_power_of_two()); assert_eq!(f[0], 1.into()); let mut inv = formal_power_series_inv(&f, gen); let mut der = vec![mod_int::ModInt::new(0); 2 * n]; for i in 1..n { der[i - 1] = f[i] * i as i64; } inv.resize(2 * n, 0.into()); let zeta = gen.pow((P::m() - 1) / n as i64 / 2); fft::fft(&mut der, zeta, 1.into()); fft::fft(&mut inv, zeta, 1.into()); let invlen = mod_int::ModInt::new(2 * n as i64).inv(); for i in 0..2 * n { der[i] *= inv[i] * invlen; } fft::inv_fft(&mut der, zeta.inv(), 1.into()); // integral of f'/f let mut ans = vec![mod_int::ModInt::new(0); n]; for i in 1..n { ans[i] = der[i - 1] * invfac[i] * fac[i - 1]; } ans } // Computes exp(f) mod x^{f.len()}. // Reference: https://arxiv.org/pdf/1301.5804.pdf // Complexity: O(n log n) // Depends on: ModInt.rs, fact_init.rs, fft.rs fn formal_power_series_exp( h: &[mod_int::ModInt

], gen: mod_int::ModInt

, fac: &[mod_int::ModInt

], invfac: &[mod_int::ModInt

], ) -> Vec> { let n = h.len(); assert!(n.is_power_of_two()); assert_eq!(h[0], 0.into()); let mut m = 1; let mut f = vec![mod_int::ModInt::new(0); n]; let mut g = vec![mod_int::ModInt::new(0); n]; let mut tmp_f = vec![mod_int::ModInt::new(0); n]; let mut tmp_g = vec![mod_int::ModInt::new(0); n]; let mut tmp = vec![mod_int::ModInt::new(0); n]; f[0] = 1.into(); g[0] = 1.into(); // Adopts the technique used in https://judge.yosupo.jp/submission/3153 while m < n { // upheld invariants: f = exp(h) (mod x^m) // g = exp(-h) (mod x^(m/2)) // Complexity: 4 * fft(2 * m) + 2 * fft(m) + 2 * inv_fft(2 * m) + 3 * inv_fft(m) // ~= 8.5 * fft(2 * m) let zeta2m = gen.pow((P::m() - 1) / m as i64 / 2); let zeta = zeta2m * zeta2m; // 2.a': g = 2g - fg^2 mod x^m let factor2m = mod_int::ModInt::new(m as i64 * 2).inv(); let factor = factor2m * 2; let factor2 = factor * factor; // Here we only need FFT(f[..m]), but we use it later at 2.c' tmp_f[..2 * m].copy_from_slice(&f[..2 * m]); fft::fft(&mut tmp_f[..2 * m], zeta2m, 1.into()); if m > 1 { // The following can be dropped because the actual // computation was done in the previous iteration. // tmp_g[..m].copy_from_slice(&g[..m]); // fft::fft(&mut tmp_g[..m], zeta, 1.into()); for i in 0..m { tmp[i] = tmp_f[i] * tmp_g[i]; } fft::inv_fft(&mut tmp[..m], zeta.inv(), 1.into()); for v in &mut tmp[..m / 2] { *v = 0.into(); } fft::fft(&mut tmp[..m], zeta, 1.into()); for i in 0..m { tmp[i] = -tmp[i] * tmp_g[i] * factor2; } fft::inv_fft(&mut tmp[..m], zeta.inv(), 1.into()); g[m / 2..m].copy_from_slice(&tmp[m / 2..m]); } // 2.b': q = h' mod x^(m-1) for i in 0..m - 1 { tmp[i] = h[i + 1] * (i + 1) as i64; } tmp[m - 1] = 0.into(); // 2.c': r = fq (mod x^m - 1) fft::fft(&mut tmp[..m], zeta, 1.into()); // FFT(f[..2m])[..m] == FFT(f[..m]) // Note that the result of FFT is bit-reversed. for i in 0..m { tmp[i] *= tmp_f[i] * factor; } fft::inv_fft(&mut tmp[..m], zeta.inv(), 1.into()); // 2.d' s = x(f' - r) mod (x^m - 1) for i in (0..m - 1).rev() { tmp.swap(i, i + 1); } for i in 0..m { tmp[i] = f[i] * i as i64 - tmp[i]; } // 2.e': t = gs mod x^m tmp_g[..2 * m].copy_from_slice(&g[..2 * m]); fft::fft(&mut tmp_g[..2 * m], zeta2m, 1.into()); fft::fft(&mut tmp[..2 * m], zeta2m, 1.into()); for i in 0..2 * m { tmp[i] *= tmp_g[i] * factor2m; } fft::inv_fft(&mut tmp[..2 * m], zeta2m.inv(), 1.into()); // 2.f': u = (h mod x^2m - \int tx^(m-1)) / x^m for i in 0..m { tmp[i] = h[i + m] - tmp[i] * fac[i + m - 1] * invfac[i + m]; } for v in &mut tmp[m..2 * m] { *v = 0.into(); } // 2.g': v = fu mod x^m fft::fft(&mut tmp[..2 * m], zeta2m, 1.into()); for i in 0..2 * m { tmp[i] *= tmp_f[i] * factor2m; } fft::inv_fft(&mut tmp[..2 * m], zeta2m.inv(), 1.into()); // 2.h': f += vx^m f[m..2 * m].copy_from_slice(&tmp[..m]); // 2.i': m *= 2 m *= 2; } f } fn main() { // In order to avoid potential stack overflow, spawn a new thread. let stack_size = 104_857_600; // 100 MB let thd = std::thread::Builder::new().stack_size(stack_size); thd.spawn(|| solve()).unwrap().join().unwrap(); } fn calc(v: usize, c: i64, fac: &[MInt], invfac: &[MInt]) -> Vec { // f(x) := 1 + x + x^2/2 + ... + x^lim/lim! where lim = floor(v / 2) // The value we want is (v - 1)![x^{v - 1}] (f(x)^c - f(x)^{c - 1}x^lim) + 1 let lim = v / 2; let mut p = 1; while p < v { p *= 2; } let mut dp = vec![MInt::new(0); 2 * p]; for i in 0..lim + 1 { dp[i] += invfac[i]; } let mut mul = dp.clone(); dp[lim] = 0.into(); let gen = MInt::new(3); let mut ans = vec![MInt::new(0); c as usize + 1]; let zeta = gen.pow((MOD - 1) / (2 * p) as i64); let factor = MInt::new((2 * p) as i64).inv(); fft::fft(&mut mul, zeta, 1.into()); for i in 1..c + 1 { ans[i as usize] = dp[v - 1] * fac[v - 1] + 1; if i == c { break; } fft::fft(&mut dp, zeta, 1.into()); for i in 0..2 * p { dp[i] *= mul[i] * factor; } fft::inv_fft(&mut dp, zeta.inv(), 1.into()); for j in p..2 * p { dp[j] = 0.into(); } } ans } fn solve() { input! { n: usize, c: i64, ab: [(usize1, usize1); n - 1], } let (fac, invfac) = fact_init(4 * max(n, 256) + 1); let mut deg = vec![0; n]; for &(a, b) in &ab { deg[a] += 1; deg[b] += 1; } let mut freq = vec![0; n]; for i in 0..n { freq[deg[i]] += 1; } freq[1] -= 1; let mut dp = vec![MInt::new(0); c as usize + 1]; for d in 1..c + 1 { dp[d as usize] = MInt::new(d); } for i in 1..n { for _ in 0..freq[i] { let val = calc(i, c, &fac, &invfac); for j in 1..c as usize + 1 { dp[j] *= val[j]; } } } let mut ans = MInt::new(0); for i in 1..c + 1 { let tmp = dp[i as usize] * fac[c as usize] * invfac[(c - i) as usize] * invfac[i as usize]; if (i + c) % 2 == 0 { ans += tmp; } else { ans -= tmp; } } println!("{}", ans); }