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 JavaScript (judge verdict: Wrong Answer)

class Tree {
  static get Node() {
    return class {
      constructor(id, parentId, childrenIds, depth) {
        this.id = id;
        this.parentId = parentId;
        this.childrenIds = childrenIds;
        this.depth = depth;
      }

      get type() {
        if (this.parentId === -1) return 'root';
        if (this.childrenIds.length === 0) return 'leaf';
        return 'internal node';
      }

      toString() {
        return `node ${this.id}: parent = ${this.parentId}, depth = ${this.depth}, ${this.type}, [${this.childrenIds.join(', ')}]`;
      }
    };
  }

  constructor() {
    this.nodes = [];
  }

  add(id, childrenIds) {
    this.nodes[id] = this.nodes[id] || new Tree.Node(id, -1, childrenIds, 0);
    let node = this.nodes[id];
    node.childrenIds = childrenIds;

    let c;
    if (childrenIds.length !== 0) {
      childrenIds.forEach((cid) => {
        c = this.nodes[cid] || new Tree.Node(cid, id, [], node.depth);
        c.parentId = id;
        c.depth = node.depth;
        this.nodes[cid] = c;
        this.incrementDepth(cid);
      });
    }
  }

  incrementDepth(id) {
    if (!this.nodes[id]) return;
    if (this.nodes[id].childrenIds.length === 0) {
      this.nodes[id].depth += 1;
      return;
    }
    this.nodes[id].childrenIds.forEach((eid) => this.incrementDepth(eid));
  }

  print() {
    console.log(this.nodes.sort((a, b) => a.id - b.id).map(n => n.toString()).join('\n'));
  }
}

(function main(){
  const lines = require('fs').readFileSync('/dev/stdin', 'utf8').trim().split('\n');
  const n = Number(lines.shift());

  let tree = new Tree();

  let id, k, childrenIds;
  lines.map(l => l.split(' ')).forEach(function(line) {
    id = Number(line.shift());
    k = Number(line.shift());
    childrenIds = line.map(Number);
    tree.add(id, childrenIds);
  });

  tree.print();
})();