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: Accepted)

class Node
  attr_reader :id, :left, :right
  attr_accessor :parent
  protected :parent=

  def initialize(id)
    @id = id
  end

  def left=(node)
    @left = node
    node.parent = self
  end

  def right=(node)
    @right = node
    node.parent = self
  end

  def grand_parent
    @parent.nil? ? self : @parent.grand_parent
  end
end

class NodeRepository
  include Enumerable

  def initialize
    @store = {}
  end

  def [](id)
    @store[id] ||= Node.new(id)
  end

  # The order may be random.
  def each(&block)
    Enumerator.new do |y|
      @store.each { |_, v| y << v }
    end.each(&block)
  end
end
repo = NodeRepository.new

n = gets.chomp.to_i
n.times do
  current, left, right = gets.chomp.split(' ').map(&:to_i)
  node = repo[current]
  node.left = repo[left] if left != -1
  node.right = repo[right] if right != -1
end
root = repo.first.grand_parent

class TreeWalker
  def self.enum_node_preorder(node, &block)
    yield node
    enum_node_preorder(node.left, &block) if node.left
    enum_node_preorder(node.right, &block) if node.right
  end
  def self.enum_node_inorder(node, &block)
    enum_node_inorder(node.left, &block) if node.left
    yield node
    enum_node_inorder(node.right, &block) if node.right
  end
  def self.enum_node_postorder(node, &block)
    enum_node_postorder(node.left, &block) if node.left
    enum_node_postorder(node.right, &block) if node.right
    yield node
  end

  def initialize(root)
    @root = root
  end

  def enum_preorder(&block)
    self.class.enum_node_preorder(@root, &block)
  end
  def enum_inorder(&block)
    self.class.enum_node_inorder(@root, &block)
  end
  def enum_postorder(&block)
    self.class.enum_node_postorder(@root, &block)
  end
end
walker = TreeWalker.new(root)

puts "Preorder"
print ' '
puts walker.to_enum(:enum_preorder).map(&:id).join(' ')

puts "Inorder"
print ' '
puts walker.to_enum(:enum_inorder).map(&:id).join(' ')

puts "Postorder"
print ' '
puts walker.to_enum(:enum_postorder).map(&:id).join(' ')