Problem:
Tree Walk

Binary trees are defined recursively. A binary tree T is a structure defined on a finite set of nodes that either

contains no nodes, or

is composed of three disjoint sets of nodes:

- a root node.

- a binary tree called its left subtree.

- a binary tree called its right subtree.

Your task is to write a program which perform tree walks (systematically traverse all nodes in a tree) based on the following algorithms:

Print the root, the left subtree and right subtree (preorder).

Print the left subtree, the root and right subtree (inorder).

Print the left subtree, right subtree and the root (postorder).

Here, the given binary tree consists of n nodes and evey node has a unique ID from 0 to n-1.

Input

The first line of the input includes an integer n, the number of nodes of the tree.

In the next n linen, the information of each node is given in the following format:

id left right

id is the node ID, left is ID of the left child and right is ID of the right child. If the node does not have the left (right) child, the left(right) is indicated by -1

Output

In the 1st line, print "Preorder", and in the 2nd line print a list of node IDs obtained by the preorder tree walk.

In the 3rd line, print "Inorder", and in the 4th line print a list of node IDs obtained by the inorder tree walk.

In the 5th line, print "Postorder", and in the 6th line print a list of node IDs obtained by the postorder tree walk.

Print a space character before each node ID.

Constraints

1 ≤ n ≤ 25

Sample Input 1

9
0 1 4
1 2 3
2 -1 -1
3 -1 -1
4 5 8
5 6 7
6 -1 -1
7 -1 -1
8 -1 -1

Sample Output 1

Preorder
0 1 2 3 4 5 6 7 8
Inorder
2 1 3 0 6 5 7 4 8
Postorder
2 3 1 6 7 5 8 4 0

Reference

Introduction to Algorithms, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The MIT Press.

Solution in Ruby (judge verdict: Wrong Answer)

class Node
  attr_accessor :id, :left, :right, :parent
  def initialize(input)
    @id, @left, @right = input
    @parent = nil
  end
end

class Binary_tree
  def initialize(n, inputs)
    #nodesを生成
    @nodes = inputs.inject(Hash.new) do |memo, input|
      memo[input[0]] = Node.new(input)
      memo
    end

    #left, rightにポインタを代入
    @nodes.each do |key, node|
      @nodes[key].left = (@nodes[key].left != -1 ? @nodes[@nodes[key].left] : nil)
      @nodes[key].right = (@nodes[key].left != -1 ? @nodes[@nodes[key].right] : nil)
    end

    @root = get_root()
  end

  def get_root
    @nodes.each do |key, node|
      if !@nodes[key].left || !@nodes[key].right
        next
      end

      @nodes[key].left.parent = @nodes[@nodes[key]]
      @nodes[key].right.parent = @nodes[@nodes[key]]
    end

    @nodes.each do |key, node|
      if !node.parent
        return key
      end
    end
  end


  def preorder(node)
    if node == nil
      return
    end
    print " %d" % [node.id.to_s]
    preorder(node.left)
    preorder(node.right)
  end

  def inorder(node)
    if node == nil
      return
    end
    inorder(node.left)
    print " %d" % [node.id.to_s]
    inorder(node.right)
  end

  def postorder(node)
    if node == nil
      return
    end
    postorder(node.left)
    postorder(node.right)
    print " %d" % [node.id.to_s]
  end

  def run
    puts "Preorder"
    preorder(@nodes[@root])
    print "\n"

    puts "Inorder"
    inorder(@nodes[@root])
    print "\n"

    puts "Postorder"
    postorder(@nodes[@root])
    print "\n"
  end
end

n = gets.to_i
inputs = Array.new(n){gets.split.map(&:to_i)}
Binary_tree.new(n, inputs).run