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)

$a=[]
rootID=0
class Tree
    attr_accessor :id, :parent, :sibling, :degree, :depth, :height, :children, :type, :inPrinted
    def initialize(id, children, degree)
        @id = id
        @parent = -1
        @sibling = -1
        @degree = degree
        @height = 0
        @children = children
        @type = (children == [-1, -1] ? 'leaf' : 'internal node')
        @inrinted= false
    end

    def setDepth
        children.each {|c|
            unless c == -1
                $a[c].depth = depth + 1
                $a[c].setDepth
            end
        }
    end

    def setHeight
        unless parent == -1
            $a[parent].height = height + 1 if $a[parent].height < height + 1
            $a[parent].setHeight
        end
    end

    def data
        puts "node #{id}: parent = #{parent}, sibling = #{sibling}, degree = #{degree}, depth = #{depth}, height = #{height}, #{type}"
    end

    def preorder
        print ' ', id
        $a[children[0]].preorder unless children[0] == -1
        $a[children[1]].preorder unless children[1] == -1
    end

    def inorder
        $a[children[0]].inorder unless children[0] == -1
        print ' ', id
        $a[children[1]].inorder unless children[1] == -1
    end

    def postorder
        $a[children[0]].postorder unless children[0] == -1
        $a[children[1]].postorder unless children[1] == -1
        print ' ', id
    end
end

N = gets.to_i

N.times {
    id, *children = gets.split.map(&:to_i)
    degree = 2
    children.each {|e| degree -= 1 if e == -1}
    $a << Tree.new(id, children, degree)
}
$a.sort_by! {|e| e.id}
$a.each {|ele|
    ele.children.each {|e|
        unless e == -1
            $a[e].parent = ele.id
            $a[e].sibling = (e == ele.children[0] ? ele.children[1] : ele.children[0])
        end
    }
}
$a.each {|e|
    if e.parent == -1
        e.type = 'root'
        e.depth = 0
        e.setDepth
        rootID = e.id
    end

    e.setHeight if e.type == 'leaf'
}
$a.each {|e| e.data}
puts 'Preorder'
$a[rootID].preorder
puts '', 'Inorder'
$a[rootID].inorder
puts '', 'Postorder'
$a[rootID].postorder
puts