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: Runtime Error)

#include<stdio.h>
#define NUM 100000

typedef struct Node{
  int parent,left,right;
}Node;

Node T[NUM];
int n,A[NUM],B[NUM];

void depth(int a, int b){
  if(a == -1)
    return;

  A[a] = b;
  depth(T[a].left, b+1);
  depth(T[a].right, b+1);
}

int height(int a){
  int h1=0,h2=0;

  if(T[a].left != -1)
    h1 = height(T[a].left)+1;

  if(T[a].right != -1)
    h2 = height(T[a].right)+1;

  if(h1 > h2)
    return B[a] = h1;

  else
    return B[a] = h2;
}

int sibling(int a){
  if(T[a].parent == -1)
    return -1;

  if(T[T[a].parent].left != a && T[T[a].parent].left != -1)
    return T[T[a].parent].left;

  if(T[T[a].parent].right != a && T[T[a].parent].right != -1)
    return T[T[a].parent].right;

  return -1;
}

void print(int a){
  int d = 0;

  printf("node %d: parent = %d, sibling = %d, ", a, T[a].parent, sibling(a));

  if(T[a].left != -1)
    d++; 

  if(T[a].right != -1)
    d++;

  printf("degree = %d, depth = %d, height = %d, ", d, A[a], B[a]);


  if(T[a].parent == -1)
    printf("root\n");

  else if(T[a].left == -1 && T[a].right == -1)
    printf("leaf\n");

  else
    printf("internal node\n");
}

int main(){
  int a,l,ri,ro,i;

  scanf("%d", &n);

  for(i = 0; i < n; i++)
    T[i].parent = -1;

  for(i = 0;i < n; i++){
    scanf("%d%d%d", &a, &l, &ri);

    T[a].left = l;
    T[a].right = ri;

    if(l != -1)
      T[l].parent = a;

    if(ri != -1)
      T[ri].parent = a;
  }

  for(i = 0; i < n; i++){
    if(T[i].parent == -1)
      ro = i;
  }

  depth(ro, 0);
  height(ro);

  for(i = 0; i < n; i++)
    print(i);

  return 0;
}