Problem:
Rooted Trees

A graph G = (V, E) is a data structure where V is a finite set of vertices and E is a binary relation on V represented by a set of edges. Fig. 1 illustrates an example of a graph (or graphs).

Fig. 1

A free tree is a connnected, acyclic, undirected graph. A rooted tree is a free tree in which one of the vertices is distinguished from the others. A vertex of a rooted tree is called "node."

Your task is to write a program which reports the following information for each node u of a given rooted tree T:

node ID of u

parent of u

depth of u

node type (root, internal node or leaf)

a list of chidlren of u

If the last edge on the path from the root r of a tree T to a node x is (p, x), then p is the parent of x, and x is a child of p. The root is the only node in T with no parent.

A node with no children is an external node or leaf. A nonleaf node is an internal node

The number of children of a node x in a rooted tree T is called the degree of x.

The length of the path from the root r to a node x is the depth of x in T.

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

Fig. 2 shows an example of rooted trees where ID of each node is indicated by a number in a circle (node). The example corresponds to the first sample input.

Fig. 2

Input

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

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

id k c1 c2 ... ck

where id is the node ID of u, k is the degree of u, c1 ... ck are node IDs of 1st, ... kth child of u. If the node does not have a child, the k is 0.

Output

Print the information of each node in the following format ordered by IDs:

node id: parent = p, depth = d, type, [c1...ck]

p is ID of its parent. If the node does not have a parent, print -1.

d is depth of the node.

type is a type of nodes represented by a string (root, internal node or leaf). If the root can be considered as a leaf or an internal node, print root.

c1...ck is the list of children as a ordered tree.

Please follow the format presented in a sample output below.

Constraints

1 ≤ n ≤ 100000

Sample Input 1

13
0 3 1 4 10
1 2 2 3
2 0
3 0
4 3 5 6 7
5 0
6 0
7 2 8 9
8 0
9 0
10 2 11 12
11 0
12 0

Sample Output 1

node 0: parent = -1, depth = 0, root, [1, 4, 10]
node 1: parent = 0, depth = 1, internal node, [2, 3]
node 2: parent = 1, depth = 2, leaf, []
node 3: parent = 1, depth = 2, leaf, []
node 4: parent = 0, depth = 1, internal node, [5, 6, 7]
node 5: parent = 4, depth = 2, leaf, []
node 6: parent = 4, depth = 2, leaf, []
node 7: parent = 4, depth = 2, internal node, [8, 9]
node 8: parent = 7, depth = 3, leaf, []
node 9: parent = 7, depth = 3, leaf, []
node 10: parent = 0, depth = 1, internal node, [11, 12]
node 11: parent = 10, depth = 2, leaf, []
node 12: parent = 10, depth = 2, leaf, []

Sample Input 2

4
1 3 3 2 0
0 0
3 0
2 0

Sample Output 2

node 0: parent = 1, depth = 1, leaf, []
node 1: parent = -1, depth = 0, root, [3, 2, 0]
node 2: parent = 1, depth = 1, leaf, []
node 3: parent = 1, depth = 1, leaf, []

Note

You can use a left-child, right-sibling representation to implement a tree which has the following data:

the parent of u

the leftmost child of u

the immediate right sibling of u

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)

# frozen_string_literal: true
if !ENV['RUBY_THREAD_VM_STACK_SIZE']
  #Rubyパスを取得するには、rbconfigかrubygemsを使う。AtCoderでは--disable-gemsされているので、require 'rubygems'は必須である。
  #require 'rbconfig';RUBY=File.join(RbConfig::CONFIG['bindir'],RbConfig::CONFIG['ruby_install_name'])
  require 'rubygems';RUBY=Gem.ruby
  exec({'RUBY_THREAD_VM_STACK_SIZE'=>'100000000'},RUBY,$0) #100MB
end

class Node
  attr_accessor :parent, :left, :right, :depth

  def initialize(parent, left, right)
    @parent = parent || -1
    @left = left
    @right = right
    @depth = nil
  end
end

def set_depth(node, p, nodes)
  node.depth = p
  set_depth(nodes[node.right], p, nodes) if node.right
  set_depth(nodes[node.left], p + 1, nodes) if node.left
end

def children(node, nodes)
  children = []
  child = node.left
  while child
    children << child
    child = nodes[child].right
  end

  children
end

n = gets.to_i
array = readlines.map { |s| s.split.map(&:to_i) }

nodes = []
n.times do
  nodes << Node.new(nil, nil, nil)
end

array.each do |e|
  id = e[0]
  k = e[1]
  children = e.slice(2..-1)

  nodes[id].left = children.first
  children.each_with_index do |child, i|
    nodes[child].parent = id
    nodes[child].right = children[i + 1]
  end
end

root = nodes.find { |e| e.parent == -1 }

set_depth(root, 0, nodes)

nodes.each_with_index do |node, id|
  type = if node.parent == -1
           'root'
         elsif node.left
           'internal node'
         else
           'leaf'
  end

  puts "node #{id}: parent = #{node.parent}, depth = #{node.depth}, #{type}, #{children(node, nodes).inspect}"
end