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 <string.h>
#include <stdlib.h>

#define MAX_N 100000

struct Node{
  int key;
  struct Node *next;
  struct Node *prev;
};

typedef struct Node* NodePointer;
NodePointer nil[MAX_N];

int memo[MAX_N];
int getDepth(int,int*);
void init(int);
void insert(int,int);
void displayList(int,int);

int main(){
  int n,i,j,id,child,type;
  int parent[MAX_N],k[MAX_N],num[MAX_N];
  char name[3][14] = {"root","leaf","internal node"};

  memset(parent,-1,sizeof(parent));
  memset(memo,-1,sizeof(memo));
  memset(num,0,sizeof(num));

  scanf("%d",&n);
  for(i = 0 ; i < n ; i++){
    scanf("%d%d",&id,&k[i]); init(id);
    num[id] = k[i];
    for(j = 0 ; j < k[i] ; j++){
      scanf("%d",&child);
      parent[child] = id;
      insert(child,id);
    }
  }
  for(i = 0 ; i < n ; i++){
    printf("node %d: parent = %d, depth = %d, "
           ,i,parent[i],getDepth(i,parent));
    if(parent[i] == -1){ type = 0; }
    else if(num[i] == 0){ type = 1; }
    else{ type = 2; }
    printf("%s, [",name[type]);
    displayList(i,k[i]);
  }
  return 0;
}

int getDepth(int x,int *parent){
  if(parent[x] == -1){ return 0; }
  if(memo[x] >= 0){ return memo[x]; }
  return memo[x] = getDepth(parent[x],parent) + 1;
}

void init(int idx){
  nil[idx] = malloc(sizeof(struct Node));
  nil[idx]->next = nil[idx];
  nil[idx]->prev = nil[idx];
  nil[idx]->key = -1;
}

void insert(int key,int idx){
  NodePointer x,y;
  x = malloc(sizeof(struct Node));
  x->key = key;
   
  y = nil[idx]->next;
  nil[idx]->next = x;
  x->prev = nil[idx];
  x->next = y;
  y->prev = x;
}

void displayList(int idx,int len){
  NodePointer cur;
  int i = 0;
  for(cur = nil[idx]->prev ; cur != nil[idx] ; cur = cur->prev, i++){
    if(i > 0){ printf(", "); }
    printf("%d",cur->key);
  }
  printf("]\n");
}