Problem:
Max Score: $800$ Points

Problem Statement

There is an undirected connected graph with $N$ vertices and $N-1$ edges. The i-th edge connects u_i and v_i.

E869120 the coder moves in the graph as follows:

He move to adjacent vertex, but he can't a visit vertex two or more times.

He ends move when there is no way to move.

Otherwise, he moves randomly. (equal probability) If he has $p$ way to move this turn, he choose each vertex with $1/p$ probability.

Calculate the expected value of the number of turns, when E869120 starts from vertex i, for all i (1 ≤ i ≤ N).

Input

The input is given from standard input in the following format.

N
u_1 v_1
u_2 v_2
:
u_{N-1} v_{N-1}

Output

In i-th (1 ≤ i ≤ N) line, print the expected vaule of the number of turns E869120 moves.

The relative error or absolute error of output should be within 10^{-6}.

Constraints

$1 \le N \le 150,000$

The graph is connected.

Subtasks

Subtask 1 [ $190$ points ]

There is no vertex which degree is more than 2.

This means the graph looks like a list.

Subtask 2 [ $220$ points ]

1 ≤ N ≤ 1000.

Subtask 3 [ $390$ points ]

There are no additional constraints.

Sample Input 1

4
1 2
2 3
2 4

Sample Output 1

2.0
1.0
2.0
2.0

Sample Input 2

4
1 2
2 4
4 3

Solution in Common Lisp (SBCL 2.0.3) (judge verdict: Accepted)

#-swank
(unless (member :child-sbcl *features*)
  (quit
   :unix-status
   (process-exit-code
    (run-program *runtime-pathname*
                 `("--control-stack-size" "128MB"
                   "--noinform" "--disable-ldb" "--lose-on-corruption" "--end-runtime-options"
                   "--eval" "(push :child-sbcl *features*)"
                   "--script" ,(namestring *load-pathname*))
                 :output t :error t :input t))))
(in-package :cl-user)
(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-int:defconstant-eqx opt
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0))
    #'equal)
  #+swank (ql:quickload '(:cl-debug-print :fiveam) :silent t)
  #-swank (set-dispatch-macro-character
           #\# #\> (lambda (s c p) (declare (ignore c p)) `(values ,(read s nil nil t)))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

(defconstant +mod+ 1000000007)

(defmacro dbg (&rest forms)
  #+swank (if (= (length forms) 1)
              `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
              `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

;; BEGIN_INSERTED_CONTENTS
(defpackage :cp/read-fixnum
  (:use :cl)
  (:export #:read-fixnum))
(in-package :cp/read-fixnum)

(declaim (ftype (function * (values fixnum &optional)) read-fixnum))
(defun read-fixnum (&optional (in *standard-input*))
  "NOTE: cannot read -2^62"
  (macrolet ((%read-byte ()
               `(the (unsigned-byte 8)
                     #+swank (char-code (read-char in nil #\Nul))
                     #-swank (sb-impl::ansi-stream-read-byte in nil #.(char-code #\Nul) nil))))
    (let* ((minus nil)
           (result (loop (let ((byte (%read-byte)))
                           (cond ((<= 48 byte 57)
                                  (return (- byte 48)))
                                 ((zerop byte) ; #\Nul
                                  (error "Read EOF or #\Nul."))
                                 ((= byte #.(char-code #\-))
                                  (setq minus t)))))))
      (declare ((integer 0 #.most-positive-fixnum) result))
      (loop
        (let* ((byte (%read-byte)))
          (if (<= 48 byte 57)
              (setq result (+ (- byte 48)
                              (* 10 (the (integer 0 #.(floor most-positive-fixnum 10))
                                         result))))
              (return (if minus (- result) result))))))))

;; BEGIN_USE_PACKAGE
(eval-when (:compile-toplevel :load-toplevel :execute)
  (use-package :cp/read-fixnum :cl-user))
(in-package :cl-user)

;;;
;;; Body
;;;

(defun main ()
  (let* ((n (read))
         (graph (make-array n :element-type 'list :initial-element nil))
         (dp (make-array n :element-type 'double-float :initial-element 0d0))
         (res (make-array n :element-type 'double-float :initial-element 0d0))
         (degs (make-array n :element-type 'uint31 :initial-element 0)))
    (dotimes (i (- n 1))
      (let ((u (- (read-fixnum) 1))
            (v (- (read-fixnum) 1)))
        (incf (aref degs u))
        (incf (aref degs v))
        (push u (aref graph v))
        (push v (aref graph u))))
    (sb-int:named-let dfs ((v 0) (parent -1))
      (let ((sum 0d0)
            (count (+ (aref degs v) (if (= -1 parent) 0 -1))))
        (dolist (child (aref graph v))
          (unless (= child parent)
            (dfs child v)
            (incf sum (+ 1 (aref dp child)))))
        (setf (aref dp v)
              (if (zerop count) 0d0 (/ sum count)))))
    #>dp
    (sb-int:named-let dfs ((v 0) (parent -1))
      (setf (aref res v) (aref dp v))
      (let ((v-deg (aref degs v)))
        (dolist (child (aref graph v))
          (unless (= child parent)
            (let* ((old-v-value (aref dp v))
                   (old-child-value (aref dp child))
                   (new-v-value (let ((num (* (- old-v-value
                                                 (/ (+ old-child-value 1) v-deg))
                                              v-deg)))
                                  (if (zerop num)
                                      num
                                      (/ num (- v-deg 1)))))
                   (new-child-value (/ (+ (* (- (aref degs child) 1) old-child-value)
                                          new-v-value
                                          1)
                                       (aref degs child))))
              (setf (aref dp v) new-v-value
                    (aref dp child) new-child-value)
              (dfs child v)
              (setf (aref dp v) old-v-value
                    (aref dp child) old-child-value))))))
    (write-string
     (with-output-to-string (*standard-output* nil :element-type 'base-char)
       (map () #'println res)))))

#-swank (main)

;;;
;;; Test and benchmark
;;;

#+swank
(defun get-clipbrd ()
  (with-output-to-string (out)
    #+os-windows (run-program "powershell.exe" '("-Command" "Get-Clipboard") :output out :search t)
    #+os-unix (run-program "xsel" '("-b" "-o") :output out :search t)))

#+swank (defparameter *this-pathname* (uiop:current-lisp-file-pathname))
#+swank (defparameter *dat-pathname* (uiop:merge-pathnames* "test.dat" *this-pathname*))

#+swank
(defun run (&optional thing (out *standard-output*))
  "THING := null | string | symbol | pathname

null: run #'MAIN using the text on clipboard as input.
string: run #'MAIN using the string as input.
symbol: alias of FIVEAM:RUN!.
pathname: run #'MAIN using the text file as input."
  (let* ((*standard-output* (or out (make-string-output-stream)))
         (res (etypecase thing
                (null
                 (with-input-from-string (*standard-input* (delete #\Return (get-clipbrd)))
                   (main)))
                (string
                 (with-input-from-string (*standard-input* (delete #\Return thing))
                   (main)))
                (symbol (5am:run! thing))
                (pathname
                 (with-open-file (*standard-input* thing)
                   (main))))))
    (if out res (get-output-stream-string *standard-output*))))

#+swank
(defun gen-dat ()
  (uiop:with-output-file (out *dat-pathname* :if-exists :supersede)
    (format out "")))

#+swank
(defun bench (&optional (out (make-broadcast-stream)))
  (time (run *dat-pathname* out)))

;; To run: (5am:run! :sample)
#+swank
(it.bese.fiveam:test :sample
  (it.bese.fiveam:is
   (equal "2.0
1.0
2.0
2.0
"
          (run "4
1 2
2 3
2 4
" nil)))
  (it.bese.fiveam:is
   (equal "3.0
1.5
3.0
1.5
"
          (run "4
1 2
2 4
4 3
" nil)))
  (it.bese.fiveam:is
   (equal "4.0
2.0
2.0
2.0
4.0
"
          (run "5
1 2
2 3
3 4
4 5
" nil)))
  (it.bese.fiveam:is
   (equal "2.000000000000
1.666666666667
1.666666666667
3.000000000000
3.000000000000
3.000000000000
3.000000000000
"
          (run "7
1 2
1 3
2 4
2 5
3 6
3 7
" nil)))
  (it.bese.fiveam:is
   (equal "3.666666666667
2.250000000000
3.666666666667
2.833333333333
2.555555555556
2.666666666667
4.333333333333
2.666666666667
5.333333333333
2.500000000000
2.500000000000
5.000000000000
"
          (run "12
1 2
2 3
2 4
4 5
5 6
5 7
6 8
8 9
2 10
10 11
11 12
" nil)))
  (it.bese.fiveam:is
   (equal "1.0
1.0
"
          (run "2
1 2
" nil))))