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 C# (judge verdict: Wrong Answer)

using System;
using System.Linq;
using System.IO;
using System.Text;
using System.Collections;
using System.Collections.Generic;
 
class Program
{
    static void Main()
    {
        int n = int.Parse(Console.ReadLine());
        RootedTree rtree = new RootedTree();
        for (int i = 0; i < n; i++)
        {
            List<int> inputs = Console.ReadLine().Split(' ')
                                        .Select(val => int.Parse(val))
                                        .ToList();
            rtree.Add(inputs[0]);
            for (int j = 2; j < inputs.Count; j++)
            {
                rtree.Add(inputs[j], inputs[0]);
            }
        }
        rtree.CalcDepth();
        for (int i = 0; i < n; i++)
        {
            Console.WriteLine(rtree[i].ToString());
        }
    }
}

class RootedTree    //int??????
{
    #region class Node define
    public class Node
    {
        internal const int MaxValue = 100000;
        public int Value {get; internal set;}
        internal Node parent, leftMostChild, NearestRightSibiling;
        internal int depth;
        string Type
        {
            get
            {
                if(parent == null) return "root";
                else if(leftMostChild == null) return "leaf";
                else return "internal node";
            }
        }

        public Node(int val, Node parent)
        {
            this.Value = val;
            if(parent != null)
            {
                this.parent = parent;
                if(this.parent.leftMostChild == null) this.parent.leftMostChild = this;
                else
                {
                    Node node = this.parent.leftMostChild;
                    while (node.NearestRightSibiling != null) node = node.NearestRightSibiling;
                    node.NearestRightSibiling = this;
                }
            }
        }

        public override string ToString(){
            string result = string.Format("node {0}: parent = {1}, depth = {2}, {3}, ["
            , this.Value, this.parent != null ? this.parent.Value.ToString() : "-1", this.depth, this.Type);
            if(this.leftMostChild != null)
            {
                Node child = this.leftMostChild;
                result += child.Value;
                while(child.NearestRightSibiling != null)
                {
                    child = child.NearestRightSibiling;
                    result += ", " + child.Value;
                }
            }
            result += "]";
            return result;
        }
    }
    #endregion //class Node define

    public Node Root {get; internal set;}
    private Node[] nodes = new Node[Node.MaxValue + 1];
    public Node this[int i]
    {
        get
        {
            return nodes[i];
        }
        set
        {
            nodes[i] = value;
        }
    }

    public RootedTree()
    {
        Root = null;
    }

    public void Add(int i)
    {
        if(this[i] == null)
        {
            this[i] = new Node(i, null);
            this.Root = this[i];
        }
    }
    public void Add(int i, int iOfparent)
    {
        if(this[iOfparent] == null) Add(i);
        else
        {
            if(this[i] == null) this[i] = new Node(i, this[iOfparent]);
            else
            {
                this[i].parent = this[iOfparent];
                if(this[iOfparent].leftMostChild == null) this[iOfparent].leftMostChild = this[i];
                else
                {
                    Node node = this[iOfparent].leftMostChild;
                    while (node.NearestRightSibiling != null) node = node.NearestRightSibiling;
                    node.NearestRightSibiling = this[i];
                }
            }
        }
    }

    public void CalcDepth()
    {
        CalcDepth(this.Root, 0);
    }
    private void CalcDepth(Node node, int depth)
    {
        node.depth = depth;
        if(node.NearestRightSibiling != null) CalcDepth(node.NearestRightSibiling, depth);
        if(node.leftMostChild != null) CalcDepth(node.leftMostChild, depth + 1);
    }
}