Problem:
Score : 600 points

Problem Statement

We have a tree with N vertices numbered 1 to N. The i-th edge in this tree connects Vertex a_i and b_i.
For each k=1, ..., N, solve the problem below:

Consider writing a number on each vertex in the tree in the following manner:

First, write 1 on Vertex k.

Then, for each of the numbers 2, ..., N in this order, write the number on the vertex chosen as follows:

Choose a vertex that still does not have a number written on it and is adjacent to a vertex with a number already written on it. If there are multiple such vertices, choose one of them at random.

Find the number of ways in which we can write the numbers on the vertices, modulo (10^9+7).

Constraints

2 \leq N \leq 2 \times 10^5

1 \leq a_i,b_i \leq N

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N
a_1 b_1
:
a_{N-1} b_{N-1}

Output

For each k=1, 2, ..., N in this order, print a line containing the answer to the problem.

Sample Input 1

3
1 2
1 3

Sample Output 1

2
1
1

The graph in this input is as follows:

For k=1, there are two ways in which we can write the numbers on the vertices, as follows:

Writing 1, 2, 3 on Vertex 1, 2, 3, respectively

Writing 1, 3, 2 on Vertex 1, 2, 3, respectively

Sample Input 2

2
1 2

Sample Output 2

1
1

The graph in this input is as follows:

Sample Input 3

5
1 2
2 3
3 4
3 5

Sample Output 3

2
8
12
3
3

The graph in this input is as follows:

Sample Input 4

8
1 2
2 3
3 4
3 5
3 6
6 7
6 8

Sample Output 4

40
280
840
120
120
504
72
72

The graph in this input is as follows:

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <algorithm>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cmath>
#include <cstdint>
#include <cstdio>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <iostream>
#include <limits>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <vector>
using namespace std;
using ll = long long;
#define fst first
#define snd second

/* clang-format off */
template <class T, size_t D> struct _vec { using type = vector<typename _vec<T, D - 1>::type>; };
template <class T> struct _vec<T, 0> { using type = T; };
template <class T, size_t D> using vec = typename _vec<T, D>::type;
template <class T> vector<T> make_v(size_t size, const T& init) { return vector<T>(size, init); }
template <class... Ts> auto make_v(size_t size, Ts... rest) { return vector<decltype(make_v(rest...))>(size, make_v(rest...)); }
template <class T> inline void chmin(T &a, const T& b) { if (b < a) a = b; }
template <class T> inline void chmax(T &a, const T& b) { if (b > a) a = b; }
/* clang-format on */

template <std::uint_fast64_t Modulus>
class modint {
  using u64 = std::uint_fast64_t;

 public:
  u64 a;

  constexpr modint(const u64 x = 0) noexcept
      : a(x % Modulus) {
  }
  constexpr u64 &value() noexcept {
    return a;
  }
  constexpr const u64 &value() const noexcept {
    return a;
  }
  constexpr modint operator+(const modint rhs) const noexcept {
    return modint(*this) += rhs;
  }
  constexpr modint operator-(const modint rhs) const noexcept {
    return modint(*this) -= rhs;
  }
  constexpr modint operator*(const modint rhs) const noexcept {
    return modint(*this) *= rhs;
  }
  constexpr modint operator/(const modint rhs) const noexcept {
    return modint(*this) /= rhs;
  }
  constexpr modint &operator+=(const modint rhs) noexcept {
    a += rhs.a;
    if (a >= Modulus) {
      a -= Modulus;
    }
    return *this;
  }
  constexpr modint &operator-=(const modint rhs) noexcept {
    if (a < rhs.a) {
      a += Modulus;
    }
    a -= rhs.a;
    return *this;
  }
  constexpr modint &operator*=(const modint rhs) noexcept {
    a = a * rhs.a % Modulus;
    return *this;
  }
  constexpr modint &operator/=(const modint rhs) noexcept {
    return *this *= ~rhs;
  }
  constexpr modint power(u64 exp) const noexcept {
    modint v = 1, x = *this;
    while (exp) {
      if (exp & 1) {
        v *= x;
      }
      x *= x;
      exp >>= 1;
    }
    return v;
  }
  constexpr modint operator~() const noexcept {
    return power(Modulus - 2);
  }
};

using mint = modint<1000000007>;

template< typename Data, typename T >
struct ReRooting {
 
  struct Node {
    int to, rev;
    Data data;
  };
 
  using F1 = function< T(T, T) >;
  using F2 = function< T(T, Data) >;
 
  vector< vector< Node > > g;
  vector< vector< T > > ldp, rdp;
  vector< int > lptr, rptr;
  const F1 f1;
  const F2 f2;
  const T ident;
 
  ReRooting(int n, const F1 &f1, const F2 &f2, const T &ident) :
      g(n), ldp(n), rdp(n), lptr(n), rptr(n), f1(f1), f2(f2), ident(ident) {}
 
  void add_edge(int u, int v, const Data &d) {
    g[u].emplace_back((Node) {v, (int) g[v].size(), d});
    g[v].emplace_back((Node) {u, (int) g[u].size() - 1, d});
  }
 
  void add_edge_bi(int u, int v, const Data &d, const Data &e) {
    g[u].emplace_back((Node) {v, (int) g[v].size(), d});
    g[v].emplace_back((Node) {u, (int) g[u].size() - 1, e});
  }
 
 
  T dfs(int idx, int par) {
 
    while(lptr[idx] != par && lptr[idx] < g[idx].size()) {
      auto &e = g[idx][lptr[idx]];
      ldp[idx][lptr[idx] + 1] = f1(ldp[idx][lptr[idx]], f2(dfs(e.to, e.rev), e.data));
      ++lptr[idx];
    }
    while(rptr[idx] != par && rptr[idx] >= 0) {
      auto &e = g[idx][rptr[idx]];
      rdp[idx][rptr[idx]] = f1(rdp[idx][rptr[idx] + 1], f2(dfs(e.to, e.rev), e.data));
      --rptr[idx];
    }
    if(par < 0) return rdp[idx][0];
    return f1(ldp[idx][par], rdp[idx][par + 1]);
  }
 
  vector< T > solve() {
    for(int i = 0; i < g.size(); i++) {
      ldp[i].assign(g[i].size() + 1, ident);
      rdp[i].assign(g[i].size() + 1, ident);
      lptr[i] = 0;
      rptr[i] = (int) g[i].size() - 1;
    }
    vector< T > ret;
    for(int i = 0; i < g.size(); i++) {
      ret.push_back(dfs(i, -1));
    }
    return ret;
  }
};

vector<mint> fact, factInv;
void initFact(int N) {
  fact.resize(N + 1);
  factInv.resize(N + 1);
  fact[0] = 1;
  for (int i = 1; i <= N; i++) fact[i] = fact[i - 1] * i;
  factInv[N] = ~fact[N];
  for (int i = N - 1; i >= 0; i--) factInv[i] = factInv[i + 1] * (i + 1);
}

mint choose(int n, int k) {
  if (n < 0 || k < 0 || k > n) {
    return 0;
  } else {
    return fact[n] * factInv[k] * factInv[n - k];
  }
}

int main() {
#ifdef DEBUG
  ifstream ifs("in.txt");
  cin.rdbuf(ifs.rdbuf());
#endif
  int N;
  while (cin >> N) {
    initFact((int)2 * 1e5 + 10);
    using P = pair<int, mint>;
    auto f1 = [&](const P& a, const P& b) -> P {
      mint x = a.snd;
      mint y = b.snd;
      return {a.fst + b.fst, a.snd * b.snd * choose(a.fst + b.fst, a.fst) };
    };
    auto f2 = [&](const P& a, int d) -> P {
      return {a.fst + 1, a.snd};
    };
    ReRooting<int, P> g(N, f1, f2, { 0, mint(1) });
    for (int i = 0; i < N - 1; i++) {
      int a, b;
      cin >> a >> b;
      --a;
      --b;
      g.add_edge(a, b, 0);
    }
    for (auto& x : g.solve()) {
      cout << x.snd.value() << endl;
    }
  }
  return 0;
}