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: Accepted)

using System;
using System.Collections.Generic;
using System.Text;

class Program
{

    public void Proc()
    {
        int nodeCount = int.Parse(Console.ReadLine());
        Dictionary<int, TreeNd> dic = new Dictionary<int, TreeNd>();
        List<TreeNd> list = new List<TreeNd>();
        for (int i = 0; i < nodeCount; i++)
        {
            string[] inpt = Console.ReadLine().Split(' ');
            int id = int.Parse(inpt[0]);
            int itemCount = int.Parse(inpt[1]);

            TreeNd newItem;
            if (dic.ContainsKey(id))
            {
                newItem = dic[id];
            }
            else
            {
                newItem = new TreeNd(id);
                dic.Add(id, newItem);
                list.Add(newItem);
            }
            for(int j=0; j<itemCount; j++) {
                int subId = int.Parse(inpt[2 + j]);
                TreeNd subNd;
                if (dic.ContainsKey(subId))
                {
                    subNd = dic[subId];
                }
                else
                {
                    subNd = new TreeNd(subId);
                    dic.Add(subId, subNd);
                    list.Add(subNd);
                }
                subNd.Parent = newItem;
                newItem.SubTree.Add(subNd);
            }
        }
        list.Sort((a, b) => {
            int dir = 1;
            if (a.NodeId == b.NodeId)
            {
                return 0;
            }
            else if (a.NodeId < b.NodeId)
            {
                return dir * -1;
            }
            else
            {
                return dir;
            }
        });
        foreach (TreeNd item in list)
        {
            Console.WriteLine(item.ToString());
        }
    }

    public class TreeNd
    {
        public int NodeId = -1;
        public TreeNd Parent = null;
        public List<TreeNd> SubTree = new List<TreeNd>();

        public TreeNd(int id)
        {
            this.NodeId = id;
        }
        public string GetNodeType()
        {
            if (this.Parent == null)
            {
                return "root";
            }
            else if (SubTree.Count <= 0)
            {
                return "leaf";
            }
            else
            {
                return "internal node";
            }
        }

        private Nullable<int> DepthValue = null;
        public int Depth
        {
            get
            {
                if (this.DepthValue != null)
                {
                    return this.DepthValue.Value;
                }
                if (this.Parent == null)
                {
                    this.DepthValue = 0;
                }
                else
                {
                    this.DepthValue = this.Parent.Depth + 1;
                }
                return this.DepthValue.Value;
            }
        }
        public override string ToString()
        {
            //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, []
            StringBuilder subitems = new StringBuilder();
            subitems.Append("[");
            for (int i = 0; i < this.SubTree.Count; i++)
            {
                if (i > 0)
                {
                    subitems.Append(", ");
                }
                subitems.Append(this.SubTree[i].NodeId);
            }
            subitems.Append("]");
            string ret = "node " + this.NodeId + ": parent = " + ((this.Parent == null) ? -1 : this.Parent.NodeId) + ", depth = " + this.Depth + ", " + this.GetNodeType() + ", " + subitems.ToString();
            return ret;
        }
    }

    static void Main(string[] args)
    {
        Program prg = new Program();
        prg.Proc();
    }
}