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 Scala (2.11.7) (judge verdict: Accepted)


object Main {
  import java.io.{BufferedReader, InputStream, InputStreamReader}
  import java.util.StringTokenizer
  import scala.reflect.ClassTag

  def main(args: Array[String]): Unit = {
    val out = new java.io.PrintWriter(System.out)
    new Main(out, new InputReader(System.in)).solve()
    out.flush()
  }

  private[this] val oj = System.getenv("MY_DEBUG") == null
  def DEBUG(f: => Unit): Unit = {
    if (!oj){ f }
  }
  def debug(as: Array[Boolean]): Unit = if (!oj){ debug(as.map(x => if(x) "1" else "0").mkString) }
  def debug(as: Array[Int]): Unit = if (!oj){ debug(as.mkString(" ")) }
  def debug(as: Array[Long]): Unit =if (!oj){ debug(as.mkString(" ")) }
  def debugDim(m: Array[Array[Int]]): Unit = if (!oj){
    REP(m.length) { i =>
      debug(m(i))
    }
  }
  def debugDimFlip(m: Array[Array[Long]]): Unit = if (!oj){
    REP(m(0).length) { j =>
      REP(m.length) { i =>
        System.err.print(m(i)(j))
        System.err.print(" ")
      }
      System.err.println()
    }
  }
  def debug(s: => String): Unit = {
    if (!oj){ System.err.println(s) }
  }
  def isDebug[A](debug: => A, online: => A): A = {
    if (oj) online else debug
  }

  class InputReader(val stream: InputStream) {
    private[this] val reader = new BufferedReader(new InputStreamReader(stream), 32768)
    private[this] var tokenizer: StringTokenizer = _

    private[this] def next(): String = {
      while (tokenizer == null || !tokenizer.hasMoreTokens)
        tokenizer = new StringTokenizer(reader.readLine)
      tokenizer.nextToken
    }

    def nextInt(): Int = Integer.parseInt(next())
    def nextLong(): Long = java.lang.Long.parseLong(next())
    def nextChar(): Char = next().charAt(0)

    def ni(): Int = nextInt()
    def nl(): Long = nextLong()
    def nc(): Char = nextChar()
    def ns(): String = next()
    def ns(n: Int): Array[Char] = ns().toCharArray
    def na(n: Int, offset: Int = 0): Array[Int] = map(n)(_ => ni() + offset)
    def na2(n: Int, offset: Int = 0): (Array[Int], Array[Int]) = {
      val A1, A2 = Array.ofDim[Int](n)
      REP(n) { i =>
        A1(i) = ni() + offset
        A2(i) = ni() + offset
      }
      (A1, A2)
    }
    def nm(n: Int, m: Int): Array[Array[Int]] = {
      val A = Array.ofDim[Int](n, m)
      REP(n) { i =>
        REP(m) { j =>
          A(i)(j) = ni()
        }
      }
      A
    }
    def nal(n: Int): Array[Long] = map(n)(_ => nl())
    def nm_c(n: Int, m: Int): Array[Array[Char]] = map(n) (_ => ns(m))
  }

  def REP(n: Int, offset: Int = 0)(f: Int => Unit): Unit = {
    var i = offset
    val N = n + offset
    while(i < N) { f(i); i += 1 }
  }
  def REP_r(n: Int, offset: Int = 0)(f: Int => Unit): Unit = {
    var i = n - 1 + offset
    while(i >= offset) { f(i); i -= 1 }
  }
  def TO(from: Int, to: Int)(f: Int => Unit): Unit = {
    REP(to - from + 1, from)(f)
  }
  def map[@specialized A: ClassTag](n: Int, offset: Int = 0)(f: Int => A): Array[A] = {
    val res = Array.ofDim[A](n)
    REP(n)(i => res(i) = f(i + offset))
    res
  }

  def sumL(as: Array[Int]): Long = {
    var s = 0L
    REP(as.length)(i => s += as(i))
    s
  }
  def cumSum(as: Array[Long]): Array[Long] = {
    val cum = Array.ofDim[Long](as.length + 1)
    REP(as.length) { i =>
      cum(i + 1) = cum(i) + as(i)
    }
    cum
  }
}

object Workspace {
  import Main._
  import java.util.Arrays.sort

  import scala.collection.mutable
  import math.{abs, max, min}
  import mutable.ArrayBuffer

  def packUGraph(n: Int, from: Array[Int], to: Array[Int]): Array[Array[Int]] = {
    val t = new Array[Array[Int]](n)
    val p = new Array[Int](n)
    val m = from.length
    REP(m)(i => p(from(i)) += 1)
    REP(m)(i => p(to(i)) += 1)
    REP(n)(i => t(i) = new Array(p(i)))
    REP(m) { i =>
      p(from(i)) -= 1
      t(from(i))(p(from(i))) = to(i)
      p(to(i)) -= 1
      t(to(i))(p(to(i))) = from(i)
    }
    t
  }

  /**
   * @return (depth, parent, queue)
   */
  def traceBfs(g: Array[Array[Int]], rt: Option[Int] = Some(0)): (Array[Int], Array[Int], Array[Int]) = {
    val n = g.length
    val INF = 1e9.toInt + 10
    val q, p = Array.ofDim[Int](n)
    val d = Array.fill[Int](n)(INF)
    var cur, last = 0

    def bfs(rt: Int) {
      q(last) = rt
      last += 1
      p(rt) = -1
      d(rt) = 0

      while (cur < last) {
        val v = q(cur)
        REP(g(v).length) { i =>
          val u = g(v)(i)
          if (d(u) == INF) {
            d(u) = d(v) + 1
            p(u) = v
            q(last) = u
            last += 1
          }
        }
        cur += 1
      }
    }

    rt.map(bfs).getOrElse {
      REP(n) { v =>
        if (d(v) == INF) bfs(v)
      }
    }

    (d, p, q)
  }

  // nCk % MOD を求める。下準備で階乗Fと≡MODでの階乗の逆元Iを作る
  class Comb(N: Int, MOD: Int) {
    def powMod(x: Int, n: Int, m: Int): Int = {
      def step(x: Long, n: Int, stack: Long): Long = {
        n match {
          case 0 => stack
          case _ => step(x * x % m, n / 2, if (n % 2 == 1) stack * x % m else stack)
        }
      }
      step(x, n, 1).toInt
    }

    private[this] val F = Array.ofDim[Long](N + 1)
    F(0) = 1
    REP(N) { i =>
      F(i + 1) = F(i) * (i + 1) % MOD
    }
    private[this] val I = Array.ofDim[Long](F.length)
    I(N) = powMod(F(N).toInt, MOD - 2, MOD)

    // x! = x(x-1)!
    // x!I(x!) ≡ (x-1)!I((x-1)!)
    // I((x-1)!) ≡ I(x!) * x   MODがでかいので(x-1)!はMODと素
    REP_r(N) { i =>
      I(i) = I(i + 1) * (i + 1) % MOD
    }

    def comb(n: Int, k: Int): Long = {
      if (n < k) 0
      else F(n) * I(n - k) % MOD * I(k) % MOD
    }

    def rev(x: Int): Long = {
      I(x) * F(x - 1) % MOD
    }

    /**
     * nのグループからk回重複ありで選ぶ組み合わせ数
     * n - 1のしきりとkの○で考える
     */
    def H(n: Int, k: Int): Long = {
      comb(n + k - 1, k)
    }

    /**
     * private[this]をつかってるのでgetterが必要
     * @return (F, I)
     */
    def get: (Array[Long], Array[Long]) = (F, I)
  }

}

class Main(out: java.io.PrintWriter, sc: Main.InputReader) {
  import sc._
  import Main._
  import java.util.Arrays.sort

  import scala.collection.mutable
  import math.{abs, max, min}
  import mutable.ArrayBuffer
  import Workspace._

  // toIntとか+7とかするならvalにしろ
  final private[this] val MOD = 1000000007

  def solve(): Unit = {
    val N = ni()
    val (from, to) = na2(N - 1, -1)
    val g = packUGraph(N, from, to)
    val comb = new Comb(N, MOD)
    val (fac, inv) = comb.get
    val dpCnt = Array.ofDim[Int](N)
    val dpCmb = Array.ofDim[Long](N)

    val (_, par, queue) = traceBfs(g)
    REP_r(N) { i =>
      val v = queue(i)
      dpCnt(v) = 1
      REP(g(v).length) { j =>
        val u = g(v)(j)
        if (u != par(v)) {
          dpCnt(v) += dpCnt(u)
        }
      }

      dpCmb(v) = fac(dpCnt(v) - 1)
      REP(g(v).length) { j =>
        val u = g(v)(j)
        if (u != par(v)) {
          dpCmb(v) = dpCmb(v) *
            dpCmb(u) % MOD *
            inv(dpCnt(u)) % MOD
        }
      }
    }

    val ans = Array.ofDim[Long](N)
    val dpArg = Array.ofDim[(Long, Int)](N) // (comb, cnt)
    REP(N) { i =>
      val v = queue(i)
      val divs = ArrayBuffer[(Int, (Long, Int))]() // (node, (div, cnt))
      var a = fac(N - 1)
      REP(g(v).length) { j =>
        val u = g(v)(j)
        val c = if (u == par(v)) {
          (dpArg(v)._1 * inv(dpArg(v)._2) % MOD, dpArg(v)._2)
        } else {
          (dpCmb(u) * inv(dpCnt(u)) % MOD, dpCnt(u))
        }
        divs += u -> c
        a = a * c._1 % MOD
      }
      ans(v) = a

      val n = divs.length
      val cumL, cumR = Array.ofDim[Long](n + 2)
      cumL(0) = 1
      cumR(n + 1) = 1
      REP(n) { j =>
        cumL(j + 1) = cumL(j) * divs(j)._2._1 % MOD
      }
      REP_r(n) { j =>
        cumR(j + 1) = cumR(j + 2) * divs(j)._2._1 % MOD
      }
      REP(n) { j =>
        val u = divs(j)._1
        if (u != par(v)) {
          val cnt = N - divs(j)._2._2
          val comb = fac(cnt - 1) *
            cumL(j) % MOD *
            cumR(j + 2) % MOD
          debug(s"u:$u cnt:$cnt comb:$comb")
          dpArg(u) = (comb, cnt)
        }
      }
      debug(divs.mkString(","))
      debug(s"cumL($v)")
      debug(cumL)
      debug(s"cumR($v)")
      debug(cumR)
    }

    debug(dpArg.mkString(", "))


    REP(N) { v =>
      out.println(ans(v))
    }
  }
}