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)

#include <stdio.h>
#include <stdlib.h>

struct {
  int id;
  int num_c;
  int *list_c;
  int parent;
  int depth;
  int flag;
} typedef Tree;

int main(){
  Tree *t;
  int n, id, num_c, p, i, j;

  scanf("%d", &n);
  t = (Tree *)malloc(sizeof(Tree) * n);
  if(t == NULL){
    printf("error!\n");
    exit(-1);
  }
  for(i = 0; i < n; i++){
    scanf("%d %d", &id, &num_c);
    t[id].id = id;
    t[id].num_c = num_c;
    if(num_c != 0){
      t[id].list_c = (int *)malloc(sizeof(int) * num_c);
      if( t[id].list_c == NULL){
	printf("error!\n");
	free(t);
	exit(-1);
      }
      for(j = 0; j < num_c; j++) scanf("%d", &t[id].list_c[j]); 
    }
  }
  
  for(i = 0; i < n; i++){
    t[i].parent = -1;
    t[i].depth = 0;
    t[i].flag = 0;
  }

  for(i = 0; i < n; i++){
    for(j = 0; j < t[i].num_c; j++){
      if( i == 0) t[ t[i].list_c[j] ].flag = 1; 
      t[ t[i].list_c[j] ].parent = t[i].id;  
      t[ t[i].list_c[j] ].depth += 1; 

      p = t[i].parent;
      while(p != -1){
	t[ t[i].list_c[j] ].depth += 1; 
	p = t[p].parent;
      }
      
    }
  }
  
  for(i = 0; i < n; i++){
    if(t[i].flag == 1) t[i].depth += t[ t[i].parent ].depth;
  }

  for(i = 0; i < n; i++){
    printf("node %d: parent = %d, depth = %d, ", t[i].id, t[i].parent, t[i].depth);
    if(t[i].parent == -1) printf("root, [");
    else if(t[i].num_c != 0) printf("internal node, [");
    else printf("leaf, [");
    if(t[i].num_c != 0){
      for(j = 0; j < t[i].num_c - 1; j++) printf("%d, ", t[i].list_c[j]);
      printf("%d]\n", t[i].list_c[j]);
    }
    else printf("]\n");
  }

  return 0;
}