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 OCaml (judge verdict: Time Limit Exceeded)

let nil = -1

type tree = { mutable parent : int; mutable child : int; mutable bro : int; mutable depth : int }

let make () = { parent = nil; child = nil; bro = nil; depth = nil }

let set_depth t n =
  let rec root i =
    if i = n then nil
    else if t.(i).parent = nil then i
    else root (i + 1) in
  let rec doit i d =
    t.(i).depth <- d;
    if t.(i).bro <> nil then doit t.(i).bro d;
    if t.(i).child <> nil then doit t.(i).child (d + 1) in
  doit (root 0) 0

let print_tree t =
  let rec string_of_children i c acc =
    if c = nil then acc
    else string_of_children i t.(c).bro (acc ^ (if c <> t.(i).child then ", " else "") ^ string_of_int c) in
  let string_of_children i c = string_of_children i c "" in
  t |> Array.iteri (fun i _ ->
    "node " ^ string_of_int i ^ ": parent = " ^ string_of_int t.(i).parent
       ^ ", depth = " ^ string_of_int t.(i).depth
       ^ (if t.(i).parent = nil then ", root, ["
          else if t.(i).child = nil then ", leaf, ["
          else ", internal node, [")
       ^ string_of_children i t.(i).child
       ^ "]\n"
    |> print_string)

let () =
  let n = Scanf.scanf "%d " (fun i -> i) in
  let t = Array.init n (fun _ -> make ()) in
  for _ = 0 to n - 1 do
    let (id, k) = Scanf.scanf "%d %d " (fun id k -> (id, k)) in
    let left_child = ref 0 in
    for i = 0 to k - 1 do
      let c = Scanf.scanf "%d " (fun i -> i) in
      t.(c).parent <- id;
      if i = 0 then t.(id).child <- c
      else t.(!left_child).bro <- c;
      left_child := c;
    done;
  done;
  set_depth t n;
  print_tree t