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: Time Limit Exceeded)

#include<stdio.h>
struct Node{
  int a,b,c;
};
struct Node N[10000];
int depth(int);
int sibling(int);
int n;
int D[10000],S[10000];
int main(){
  int i,j,k,l,m,q,r,s;
  scanf("%d",&n);
  for(i=0;i<n;i++){
    N[i].a=-1;
  }
  for(i=0;i<n;i++){
    scanf("%d %d %d",&q,&r,&s);
    N[q].b=r;
    N[q].c=s;
    if(r!=-1){
      N[r].a=q;
    }
    if(s!=-1){
      N[s].a=q;
    }
  }
  for(i=0;i<n;i++){
    D[i]=depth(i);
    S[i]=0;
  }
  for(i=0;i<n;i++){
    if(N[i].b==-1 && N[i].c==-1){
      S[i]=0;
      k=N[i].a;
      l=1;
      while(1){
        if(k==-1){
          break;
        }
        if(S[k]<l){
          S[k]=l;
        }
        l++;
        k=N[k].a;
      }
    }
  }
  for(i=0;i<n;i++){
    j=0;
    printf("node %d: ",i);
    printf("parent = %d, ",N[i].a);
    printf("sibling = %d, ",sibling(i));
    if(N[i].b!=-1 || N[i].c!=-1){
      j++;
    }
    printf("degree = %d, ",j);
    printf("depth = %d, ",D[i]);
    printf("height = %d, ",S[i]);
    if(N[i].a==-1){
      printf("root\n");
    }
    else if(N[i].b==-1 && N[i].c==-1){
      printf("leaf\n");
    }
    else{
      printf("internal node\n");
    }
    }
    return 0;
}

int depth(int r){
  int c=0;
  while(1){
    r=N[r].a;
    if(r==-1){
      break;
    }
    c++;
  }
  return c;
}

int sibling(int s){
  if(N[s].a==-1){
    return -1;
  }
  if(N[N[s].a].b!=s && N[N[s].a].b!=-1){
    return N[N[s].a].b;
  }
  if(N[N[s].a].c!=s && N[N[s].a].c!=-1){
    return N[N[s].a].c;
  }
  return -1;
}