C++ Library for Competitive Programming
/*
* @title グラフ/木/重心
*
* verification-helper: IGNORE
* verification-helper: PROBLEM https://atcoder.jp/contests/arc087/tasks/arc087_d
*/
#include <iostream>
#include <vector>
#include "emthrm/graph/edge.hpp"
#include "emthrm/graph/tree/centroid.hpp"
#include "emthrm/math/modint.hpp"
int main() {
using ModInt = emthrm::MInt<1000000007>;
int n;
std::cin >> n;
std::vector<std::vector<emthrm::Edge<bool>>> graph(n);
for (int i = 0; i < n - 1; ++i) {
int x, y;
std::cin >> x >> y;
--x; --y;
graph[x].emplace_back(x, y);
graph[y].emplace_back(y, x);
}
const std::vector<int> centroids = emthrm::centroid(graph);
if (centroids.size() == 2) {
std::cout << ModInt::fact(n / 2) * ModInt::fact(n / 2) << '\n';
} else {
std::vector<int> subtree(n, 1);
const auto dfs = [&graph, &subtree](auto dfs, const int par, const int ver)
-> void {
for (const emthrm::Edge<bool>& e : graph[ver]) {
if (e.dst != par) {
dfs(dfs, ver, e.dst);
subtree[ver] += subtree[e.dst];
}
}
};
dfs(dfs, -1, centroids.front());
std::vector<int> nums;
for (const emthrm::Edge<bool>& e : graph[centroids.front()]) {
nums.emplace_back(subtree[e.dst]);
}
const int m = nums.size();
std::vector dp(m + 1, std::vector(n + 1, ModInt(0)));
dp[0][0] = 1;
for (int i = 0; i < m; ++i) {
for (int j = 0; j <= n; ++j) {
for (int k = 0; k <= nums[i] && j + k <= n; ++k) {
dp[i + 1][j + k] += dp[i][j] * ModInt::nCk(nums[i], k)
* ModInt::nCk(nums[i], k) * ModInt::fact(k);
}
}
}
ModInt ans = 0;
for (int j = 0; j <= n; ++j) {
ans += (j & 1 ? -dp[m][j] : dp[m][j]) * ModInt::fact(n - j);
}
std::cout << ans << '\n';
}
return 0;
}
#line 1 "test/graph/tree/centroid.test.cpp"
/*
* @title グラフ/木/重心
*
* verification-helper: IGNORE
* verification-helper: PROBLEM https://atcoder.jp/contests/arc087/tasks/arc087_d
*/
#include <iostream>
#include <vector>
#line 1 "include/emthrm/graph/edge.hpp"
/**
* @title 辺
*/
#ifndef EMTHRM_GRAPH_EDGE_HPP_
#define EMTHRM_GRAPH_EDGE_HPP_
#include <compare>
namespace emthrm {
template <typename CostType>
struct Edge {
CostType cost;
int src, dst;
explicit Edge(const int src, const int dst, const CostType cost = 0)
: cost(cost), src(src), dst(dst) {}
auto operator<=>(const Edge& x) const = default;
};
} // namespace emthrm
#endif // EMTHRM_GRAPH_EDGE_HPP_
#line 1 "include/emthrm/graph/tree/centroid.hpp"
#include <algorithm>
#include <ranges>
#line 7 "include/emthrm/graph/tree/centroid.hpp"
#line 1 "include/emthrm/graph/edge.hpp"
/**
* @title 辺
*/
#ifndef EMTHRM_GRAPH_EDGE_HPP_
#define EMTHRM_GRAPH_EDGE_HPP_
#include <compare>
namespace emthrm {
template <typename CostType>
struct Edge {
CostType cost;
int src, dst;
explicit Edge(const int src, const int dst, const CostType cost = 0)
: cost(cost), src(src), dst(dst) {}
auto operator<=>(const Edge& x) const = default;
};
} // namespace emthrm
#endif // EMTHRM_GRAPH_EDGE_HPP_
#line 9 "include/emthrm/graph/tree/centroid.hpp"
namespace emthrm {
template <typename CostType>
std::vector<int> centroid(
const std::vector<std::vector<Edge<CostType>>>& graph) {
const int n = graph.size();
std::vector<int> subtree(n, 1), res;
const auto dfs = [&graph, n, &subtree, &res](
auto dfs, const int par, const int ver) -> void {
bool is_centroid = true;
for (const int e : graph[ver]
| std::views::transform(&Edge<CostType>::dst)) {
if (e != par) {
dfs(dfs, ver, e);
subtree[ver] += subtree[e];
is_centroid &= subtree[e] <= n / 2;
}
}
if (is_centroid && n - subtree[ver] <= n / 2) res.emplace_back(ver);
};
dfs(dfs, -1, 0);
std::sort(res.begin(), res.end());
return res;
}
} // namespace emthrm
#line 1 "include/emthrm/math/modint.hpp"
#ifndef ARBITRARY_MODINT
# include <cassert>
#endif
#include <compare>
#line 9 "include/emthrm/math/modint.hpp"
// #include <numeric>
#include <utility>
#line 12 "include/emthrm/math/modint.hpp"
namespace emthrm {
#ifndef ARBITRARY_MODINT
template <unsigned int M>
struct MInt {
unsigned int v;
constexpr MInt() : v(0) {}
constexpr MInt(const long long x) : v(x >= 0 ? x % M : x % M + M) {}
static constexpr MInt raw(const int x) {
MInt x_;
x_.v = x;
return x_;
}
static constexpr int get_mod() { return M; }
static constexpr void set_mod(const int divisor) {
assert(std::cmp_equal(divisor, M));
}
static void init(const int x) {
inv<true>(x);
fact(x);
fact_inv(x);
}
template <bool MEMOIZES = false>
static MInt inv(const int n) {
// assert(0 <= n && n < M && std::gcd(n, M) == 1);
static std::vector<MInt> inverse{0, 1};
const int prev = inverse.size();
if (n < prev) return inverse[n];
if constexpr (MEMOIZES) {
// "n!" and "M" must be disjoint.
inverse.resize(n + 1);
for (int i = prev; i <= n; ++i) {
inverse[i] = -inverse[M % i] * raw(M / i);
}
return inverse[n];
}
int u = 1, v = 0;
for (unsigned int a = n, b = M; b;) {
const unsigned int q = a / b;
std::swap(a -= q * b, b);
std::swap(u -= q * v, v);
}
return u;
}
static MInt fact(const int n) {
static std::vector<MInt> factorial{1};
if (const int prev = factorial.size(); n >= prev) {
factorial.resize(n + 1);
for (int i = prev; i <= n; ++i) {
factorial[i] = factorial[i - 1] * i;
}
}
return factorial[n];
}
static MInt fact_inv(const int n) {
static std::vector<MInt> f_inv{1};
if (const int prev = f_inv.size(); n >= prev) {
f_inv.resize(n + 1);
f_inv[n] = inv(fact(n).v);
for (int i = n; i > prev; --i) {
f_inv[i - 1] = f_inv[i] * i;
}
}
return f_inv[n];
}
static MInt nCk(const int n, const int k) {
if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) :
fact_inv(n - k) * fact_inv(k));
}
static MInt nPk(const int n, const int k) {
return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k);
}
static MInt nHk(const int n, const int k) {
return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k));
}
static MInt large_nCk(long long n, const int k) {
if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
inv<true>(k);
MInt res = 1;
for (int i = 1; i <= k; ++i) {
res *= inv(i) * n--;
}
return res;
}
constexpr MInt pow(long long exponent) const {
MInt res = 1, tmp = *this;
for (; exponent > 0; exponent >>= 1) {
if (exponent & 1) res *= tmp;
tmp *= tmp;
}
return res;
}
constexpr MInt& operator+=(const MInt& x) {
if ((v += x.v) >= M) v -= M;
return *this;
}
constexpr MInt& operator-=(const MInt& x) {
if ((v += M - x.v) >= M) v -= M;
return *this;
}
constexpr MInt& operator*=(const MInt& x) {
v = (unsigned long long){v} * x.v % M;
return *this;
}
MInt& operator/=(const MInt& x) { return *this *= inv(x.v); }
constexpr auto operator<=>(const MInt& x) const = default;
constexpr MInt& operator++() {
if (++v == M) [[unlikely]] v = 0;
return *this;
}
constexpr MInt operator++(int) {
const MInt res = *this;
++*this;
return res;
}
constexpr MInt& operator--() {
v = (v == 0 ? M - 1 : v - 1);
return *this;
}
constexpr MInt operator--(int) {
const MInt res = *this;
--*this;
return res;
}
constexpr MInt operator+() const { return *this; }
constexpr MInt operator-() const { return raw(v ? M - v : 0); }
constexpr MInt operator+(const MInt& x) const { return MInt(*this) += x; }
constexpr MInt operator-(const MInt& x) const { return MInt(*this) -= x; }
constexpr MInt operator*(const MInt& x) const { return MInt(*this) *= x; }
MInt operator/(const MInt& x) const { return MInt(*this) /= x; }
friend std::ostream& operator<<(std::ostream& os, const MInt& x) {
return os << x.v;
}
friend std::istream& operator>>(std::istream& is, MInt& x) {
long long v;
is >> v;
x = MInt(v);
return is;
}
};
#else // ARBITRARY_MODINT
template <int ID>
struct MInt {
unsigned int v;
constexpr MInt() : v(0) {}
MInt(const long long x) : v(x >= 0 ? x % mod() : x % mod() + mod()) {}
static constexpr MInt raw(const int x) {
MInt x_;
x_.v = x;
return x_;
}
static int get_mod() { return mod(); }
static void set_mod(const unsigned int divisor) { mod() = divisor; }
static void init(const int x) {
inv<true>(x);
fact(x);
fact_inv(x);
}
template <bool MEMOIZES = false>
static MInt inv(const int n) {
// assert(0 <= n && n < mod() && std::gcd(x, mod()) == 1);
static std::vector<MInt> inverse{0, 1};
const int prev = inverse.size();
if (n < prev) return inverse[n];
if constexpr (MEMOIZES) {
// "n!" and "M" must be disjoint.
inverse.resize(n + 1);
for (int i = prev; i <= n; ++i) {
inverse[i] = -inverse[mod() % i] * raw(mod() / i);
}
return inverse[n];
}
int u = 1, v = 0;
for (unsigned int a = n, b = mod(); b;) {
const unsigned int q = a / b;
std::swap(a -= q * b, b);
std::swap(u -= q * v, v);
}
return u;
}
static MInt fact(const int n) {
static std::vector<MInt> factorial{1};
if (const int prev = factorial.size(); n >= prev) {
factorial.resize(n + 1);
for (int i = prev; i <= n; ++i) {
factorial[i] = factorial[i - 1] * i;
}
}
return factorial[n];
}
static MInt fact_inv(const int n) {
static std::vector<MInt> f_inv{1};
if (const int prev = f_inv.size(); n >= prev) {
f_inv.resize(n + 1);
f_inv[n] = inv(fact(n).v);
for (int i = n; i > prev; --i) {
f_inv[i - 1] = f_inv[i] * i;
}
}
return f_inv[n];
}
static MInt nCk(const int n, const int k) {
if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) :
fact_inv(n - k) * fact_inv(k));
}
static MInt nPk(const int n, const int k) {
return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k);
}
static MInt nHk(const int n, const int k) {
return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k));
}
static MInt large_nCk(long long n, const int k) {
if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
inv<true>(k);
MInt res = 1;
for (int i = 1; i <= k; ++i) {
res *= inv(i) * n--;
}
return res;
}
MInt pow(long long exponent) const {
MInt res = 1, tmp = *this;
for (; exponent > 0; exponent >>= 1) {
if (exponent & 1) res *= tmp;
tmp *= tmp;
}
return res;
}
MInt& operator+=(const MInt& x) {
if ((v += x.v) >= mod()) v -= mod();
return *this;
}
MInt& operator-=(const MInt& x) {
if ((v += mod() - x.v) >= mod()) v -= mod();
return *this;
}
MInt& operator*=(const MInt& x) {
v = (unsigned long long){v} * x.v % mod();
return *this;
}
MInt& operator/=(const MInt& x) { return *this *= inv(x.v); }
auto operator<=>(const MInt& x) const = default;
MInt& operator++() {
if (++v == mod()) [[unlikely]] v = 0;
return *this;
}
MInt operator++(int) {
const MInt res = *this;
++*this;
return res;
}
MInt& operator--() {
v = (v == 0 ? mod() - 1 : v - 1);
return *this;
}
MInt operator--(int) {
const MInt res = *this;
--*this;
return res;
}
MInt operator+() const { return *this; }
MInt operator-() const { return raw(v ? mod() - v : 0); }
MInt operator+(const MInt& x) const { return MInt(*this) += x; }
MInt operator-(const MInt& x) const { return MInt(*this) -= x; }
MInt operator*(const MInt& x) const { return MInt(*this) *= x; }
MInt operator/(const MInt& x) const { return MInt(*this) /= x; }
friend std::ostream& operator<<(std::ostream& os, const MInt& x) {
return os << x.v;
}
friend std::istream& operator>>(std::istream& is, MInt& x) {
long long v;
is >> v;
x = MInt(v);
return is;
}
private:
static unsigned int& mod() {
static unsigned int divisor = 0;
return divisor;
}
};
#endif // ARBITRARY_MODINT
} // namespace emthrm
#line 14 "test/graph/tree/centroid.test.cpp"
int main() {
using ModInt = emthrm::MInt<1000000007>;
int n;
std::cin >> n;
std::vector<std::vector<emthrm::Edge<bool>>> graph(n);
for (int i = 0; i < n - 1; ++i) {
int x, y;
std::cin >> x >> y;
--x; --y;
graph[x].emplace_back(x, y);
graph[y].emplace_back(y, x);
}
const std::vector<int> centroids = emthrm::centroid(graph);
if (centroids.size() == 2) {
std::cout << ModInt::fact(n / 2) * ModInt::fact(n / 2) << '\n';
} else {
std::vector<int> subtree(n, 1);
const auto dfs = [&graph, &subtree](auto dfs, const int par, const int ver)
-> void {
for (const emthrm::Edge<bool>& e : graph[ver]) {
if (e.dst != par) {
dfs(dfs, ver, e.dst);
subtree[ver] += subtree[e.dst];
}
}
};
dfs(dfs, -1, centroids.front());
std::vector<int> nums;
for (const emthrm::Edge<bool>& e : graph[centroids.front()]) {
nums.emplace_back(subtree[e.dst]);
}
const int m = nums.size();
std::vector dp(m + 1, std::vector(n + 1, ModInt(0)));
dp[0][0] = 1;
for (int i = 0; i < m; ++i) {
for (int j = 0; j <= n; ++j) {
for (int k = 0; k <= nums[i] && j + k <= n; ++k) {
dp[i + 1][j + k] += dp[i][j] * ModInt::nCk(nums[i], k)
* ModInt::nCk(nums[i], k) * ModInt::fact(k);
}
}
}
ModInt ans = 0;
for (int j = 0; j <= n; ++j) {
ans += (j & 1 ? -dp[m][j] : dp[m][j]) * ModInt::fact(n - j);
}
std::cout << ans << '\n';
}
return 0;
}