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 PHP (judge verdict: Accepted)

<?php
// solve1();
solve2();

// 左子右兄弟表現の実装


// 自力解
function solve1(){
	$n = (int)trim(fgets(STDIN));

	$list = [];
	for ($i=0; $i < $n; $i++) { 
		$node = new TreeNode($i);
		$list[$node->getId()] = $node;
	}
	for ($i=0; $i < $n; $i++) { 
		$input = explode(' ', trim(fgets(STDIN)));
		$id = (int)$input[0];
		$k = (int)$input[1];
		for ($j=0; $j < $k; $j++) { 
			$list[$id]->addChild((int)$input[$j+2]);
			$list[(int)$input[$j+2]]->setParentId($id);
		}
	}
	for ($i=0; $i < $n; $i++) { 
		if ($list[$i]->getParentId() === -1) {
			// root
			$curId = $i;
			break;
		}
	}
	// objectは参照渡しされるので、$listを参照渡しにしなくても$list内のobjectは関数内部で変更される！！
	dfs($list, $curId, 0);
	$list[$curId]->setType('root');

	foreach ($list as $v) {
		echo 'node ' . $v->getId() . ': parent = ' . $v->getParentId() . ', depth = ' . $v->getDepth() . ', '  . $v->getType() . ', [' . implode(', ', $v->getChildren()) . ']' . PHP_EOL;
	}
}

function dfs($list, $id, $depth){
	$list[$id]->setDepth($depth);
	++$depth;
	if($list[$id]->hasChild()){
		$list[$id]->setType('internal node');
		$children = $list[$id]->getChildren();
		foreach ($children as $c) {
			dfs($list, $c, $depth);
		}
	}
}

/**
 * 
 */
class TreeNode
{
	private $id;
	private $parent_id = -1;
	private $children = [];
	private $depth = 0;
	private $type = 'leaf';

	function __construct($id)
	{
		$this->id = $id;
	}

	function setParentId($p_id){
		$this->parent_id = $p_id;
	}
	function addChild($c_id){
		$this->children[] = $c_id;
	}
	function hasChild(){
		return count($this->children) > 0;
	}
	function getId(){
		return $this->id;
	}
	function getParentId(){
		return $this->parent_id;
	}
	function getChildren(){
		return $this->children;
	}
	function setDepth($d){
		$this->depth = $d;
	}
	function addDepth(){
		$this->depth += 1;
	}
	function getDepth(){
		return $this->depth;
	}
	function setType($t){
		$this->type = $t;
	}
	function getType(){
		return $this->type;
	}
}


function solve2(){
	$n = (int)trim(fgets(STDIN));

	$list = [];
	for ($i=0; $i < $n; $i++) { 
		$node = new LcRsNode($i);
		$list[$i] = $node;
	}
	for ($i=0; $i < $n; $i++) { 
		$input = explode(' ', trim(fgets(STDIN)));
		$id = (int)$input[0];
		$k = (int)$input[1];
		if ($k > 0) {
			$list[$id]->setLeft((int)$input[2]);
			for ($j=0; $j < $k; $j++) { 
				$list[(int)$input[$j+2]]->setParentId($id);
				if (isset($input[$j+3])) {
					$list[(int)$input[$j+2]]->setRight((int)$input[$j+3]);
				}
			}
		}
	}
	for ($i=0; $i < $n; $i++) { 
		if ($list[$i]->getParentId() === -1) {
			// root
			$root_id = $i;
		}
	}
	dfs2($list, $root_id, 0);
	foreach ($list as $v) {
		// var_dump($v);
		$type = 'leaf';
		if ($v->getParentId() === -1) {
			$type = 'root';
		}elseif ($v->hasLeft()) {
			$type = 'internal node';
		}
		$children = [];
		if ($v->hasLeft()) {
			$child = $v->getLeft();
			$children[] = $child;
			while ($list[$child]->hasRight()) {
				$child = $list[$child]->getRight();
				$children[] = $child;
			}
		}
		echo 'node ' . $v->getId() . ': parent = ' . $v->getParentId() . ', depth = ' . $v->getDepth() . ', '  . $type . ', [' . implode(', ', $children) . ']' . PHP_EOL;
	}
}

function dfs2($list, $id, $depth){
	$list[$id]->setDepth($depth);
	++$depth;
	if($list[$id]->hasLeft()){
		$children = [];
		$child = $list[$id]->getLeft();
		$children[] = $child;
		while ($list[$child]->hasRight()) {
			$child = $list[$child]->getRight();
			$children[] = $child;
		}
		foreach ($children as $c) {
			dfs2($list, $c, $depth);
		}
	}

}

/**
 * 左子右兄弟表現node
 */
class LcRsNode
{
	private $id;
	private $depth = 0;
	private $parent_id = -1;
	// 一番左の子要素
	private $left = -1;
	// 右側に隣接する兄弟要素
	private $right = -1;

	function __construct($id)
	{
		$this->id = $id;
	}
	function setParentId($p_id){
		$this->parent_id = $p_id;
	}
	function setLeft($l_id){
		$this->left = $l_id;
	}
	function setRight($r_id){
		$this->right = $r_id;
	}
	function getId(){
		return $this->id;
	}
	function getParentId(){
		return $this->parent_id;
	}
	function getLeft(){
		return $this->left;
	}
	function getRight(){
		return $this->right;
	}
	function hasLeft(){
		return $this->left !== -1;
	}
	function hasRight(){
		return $this->right !== -1;
	}
	function setDepth($d){
		$this->depth = $d;
	}
	function addDepth(){
		$this->depth += 1;
	}
	function getDepth(){
		return $this->depth;
	}
}