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)

n = gets.chomp.to_i
nodes = []
n.times do
    tmp = gets.chomp.split.map(&:to_i)
    nodes[tmp[0]] = { id: tmp[0], parent: nil, depth: nil, type: (tmp.size)  < 3 ? "leaf" : "internal node", children: tmp[2, tmp.size - 2] }
end

# rootCand = [*0..(n - 1)]
# nodes.each do |n|
#     next if n[1] == 0
#     rootCand = rootCand - n.drop(2)
# end
# root = rootCand[0]

# ans = []
# nodes.each do |node|
#     id = node[0]
#     parent = nil
#     depth = nil
#     if node.size < 3 then
#         type = "leaf"
#         children = []
#     else
#         type = "internal node"
#         children = node[2, node.size - 2]
#     end
#     if id == root then
#         parent = -1
#         type = "root"
#         depth = 0
#     end
#     ans.push({ id: id, parent: parent, depth: depth, type: type, children: children })
# end

nodes.each do |node|
    next if node[:type] == "leaf"
    node[:children].each do |c|
        nodes[c][:parent] = node[:id]
    end
end

nodes.each do |node|
    if node[:parent] == nil then
        node[:parent] = -1
        node[:type] = "root"
    end
end

nodes.each do |node|
    depth = 0
    parent = node[:parent]
    until parent == -1 do
        parent = nodes[parent][:parent]
        depth += 1
    end
    node[:depth] = depth
end

nodes.each { |node| puts "node #{node[:id]}: parent = #{node[:parent]}, depth = #{node[:depth]}, #{node[:type]}, #{node[:children]}" }