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;

namespace ALDS1_7_A_RootedTrees
{
	class Program
	{
		static void Main ( string[] args )
		{
			int dataCount = int.Parse (Console.ReadLine ());
			Tree tree = new Tree (dataCount);

			for (int lp = 0; lp < dataCount; lp++)
			{
				int[] inputNum = Console.ReadLine ().Split (' ').Select (int.Parse).ToArray ();
				int nodeID = inputNum[0];
				int nodeCount = inputNum[1];

				int left = 0;
				for (int lp2 = 0; lp2 < nodeCount; lp2++)
				{
					int c = inputNum[2 + lp2];

					if (lp2 == 0)
					{
						int leftNode = c;
						tree.SetLeft (nodeID, leftNode);
					}
					else
					{
						int rightNode = c;
						tree.SetRight (left, rightNode);
					}

					left = c;
					tree.SetParent (c, nodeID);
				}
			}

			int root = tree.GetParent ();

			tree.GetDepth (root, 0);

			for (int lp = 0; lp < dataCount; lp++)
			{
				tree.PrintNode (lp);
			}

		}
	}

	class Node
	{
		public int Parent { set; get; }
		public int Left { set; get; }
		public int Right { set; get; }
	}

	class Tree
	{
		public Tree ( int capacity )
		{
			this.NodeArray = new Node[capacity];
			for (int lp = 0; lp < NodeArray.Length; lp++)
			{
				this.NodeArray[lp] = new Node ();
				NodeArray[lp].Parent = NodeArray[lp].Left = NodeArray[lp].Right = Null;
			}

			this.DepthArray = new int[capacity];
		}

		private Node[] NodeArray { set; get; }
		private int[] DepthArray { set; get; }
		private const int Null = -1;

		public void SetLeft ( int nodeNumber, int leftNumber )
		{
			NodeArray[nodeNumber].Left = leftNumber;
		}
		public void SetRight ( int nodeNumber, int rightNumber )
		{
			NodeArray[nodeNumber].Right = rightNumber;
		}
		public void SetParent ( int nodeNumber, int parentNumber )
		{
			NodeArray[nodeNumber].Parent = parentNumber;
		}

		public int GetParent ()
		{
			int index = 0;

			while (NodeArray[index].Parent != Null)
			{
				index++;
			}
			return index;
		}

		public int GetDepth ( int nodeNumber )
		{
			int count = 0;

			while (NodeArray[nodeNumber].Parent != Null)
			{
				nodeNumber = NodeArray[nodeNumber].Parent;
				count++;
			}

			return count;
		}

		//???????????¨????????????????????±???????±???????
		public int[] GetDepth ( int nodeNumber, int depth )
		{
			DepthArray[nodeNumber] = depth;

			if (NodeArray[nodeNumber].Right != Null)
			{
				GetDepth (NodeArray[nodeNumber].Right, depth);
			}
			if (NodeArray[nodeNumber].Left != Null)
			{
				GetDepth (NodeArray[nodeNumber].Left, depth + 1);
			}

			return DepthArray;
		}

		public void PrintNode ( int nodeNumber )
		{
			System.Text.StringBuilder output = new System.Text.StringBuilder (150);

			output.Append ("node " + nodeNumber + ": ");
			output.Append ("parent = " + NodeArray[nodeNumber].Parent + ", ");
			output.Append ("depth = " + DepthArray[nodeNumber] + ", ");

			if (NodeArray[nodeNumber].Parent == Null)
			{
				output.Append ("root, ");
			}
			else if (NodeArray[nodeNumber].Left == Null)
			{
				output.Append ("leaf, ");
			}
			else
			{
				output.Append ("internal node, ");
			}

			output.Append ("[");

			for (int lp = 0, node = NodeArray[nodeNumber].Left; node != Null; lp++, node = NodeArray[node].Right)
			{
				if (lp != 0)
					output.Append (", ");

				output.Append (node);
			}

			output.Append ("]");
		}
	}
}