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 Rust (judge verdict: Time Limit Exceeded)

type NodeId = usize;

#[derive(Debug)]
pub struct Node {
    parent: Option<NodeId>,
    first_child: Option<NodeId>,
    next_brother: Option<NodeId>,
    depth: u32,
}

impl Node {
    fn get_type(&self) -> String {
        if self.depth == 0 {
            return "root".to_owned();
        } else if self.first_child.is_some() {
            return "internal node".to_owned();
        } else {
            return "leaf".to_owned();
        }
    }
}

struct RootedTree {
    nodes: Vec<Node>,
}

impl RootedTree {
    fn n_nodes(&self) -> usize {
        self.nodes.len()
    }

    fn extract_children(&self, id: NodeId) -> Vec<NodeId> {
        match self.nodes[id].first_child {
            Some(id_oldest_child) => {
                let mut brothers = self.extract_younger_brothers(id_oldest_child);
                brothers.insert(0, id_oldest_child);
                brothers
            }
            None => Vec::new(),
        }
    }

    fn extract_younger_brothers(&self, id: NodeId) -> Vec<NodeId> {
        match self.nodes[id].next_brother {
            Some(id_brother) => {
                let mut brothers = self.extract_younger_brothers(id_brother);
                brothers.insert(0, id_brother);
                brothers
            }
            None => Vec::new(),
        }
    }

    fn append_node(&mut self, line: Vec<NodeId>) -> () {
        if line.len() < 3 {
            return;
        }
        let node_id = line[0];
        let degree = line[1];

        for d in 0..degree {
            let child = line[2 + d];
            self.nodes[child].parent = Some(node_id);

            if d == 0 {
                self.nodes[node_id].first_child = Some(child);
            } else {
                let oldest_brother_id = self.nodes[node_id].first_child.unwrap();
                let nearest = self.find_nearest_brother(oldest_brother_id);
                match nearest {
                    Some(brother) => self.nodes[brother].next_brother = Some(child),
                    None => panic!("could not find brother"),
                }
            }
        }
    }

    fn find_nearest_brother(&self, id_child: NodeId) -> Option<NodeId> {
        let mut current = id_child;
        loop {
            match self.nodes[current].next_brother {
                Some(brother_node_id) => current = brother_node_id,
                None => return Some(current),
            }
        }
    }

    fn fill_depth(&mut self, top: NodeId, depth: u32) {
        self.nodes[top].depth = depth;

        let child = self.nodes[top].first_child;
        if child.is_some() {
            let mut current = child.unwrap();
            loop {
                self.fill_depth(current, depth + 1);
                current = match self.nodes[current].next_brother {
                    Some(id) => id,
                    None => return,
                }
            }
        }
    }

    fn find_root(&self) -> NodeId {
        let mut current = 0 as NodeId;
        loop {
            match self.nodes[current].parent {
                Some(id) => current = id,
                None => return current,
            }
        }
    }
}

fn main() {
    let mut line = String::new();
    std::io::stdin().read_line(&mut line).ok();
    let n = line.trim().parse::<usize>().unwrap();

    let mut nodes: Vec<Node> = Vec::new();
    for _ in 0..n {
        nodes.push(Node {
            parent: None,
            first_child: None,
            next_brother: None,
            depth: 0,
        });
    }

    let mut tree = RootedTree { nodes: nodes };

    for _ in 0..n {
        let mut line = String::new();
        std::io::stdin().read_line(&mut line).ok();
        let inputs: Vec<NodeId> = line.split_whitespace()
            .map(|e| e.parse::<NodeId>().ok().unwrap())
            .collect();

        tree.append_node(inputs);
    }

    let root = tree.find_root();
    tree.fill_depth(root, 0);

    // for (id, node) in tree.nodes.iter().enumerate() {
    //     println!("{} {:?}", id, node);
    // }

    for id_node in 0..n {
        let node = &tree.nodes[id_node];
        println!(
            "node {id}: parent = {parent}, depth = {depth}, {node_type}, {children:?}",
            id = id_node,
            parent = node.parent.map(|x| x as i32).unwrap_or(-1),
            depth = node.depth,
            node_type = node.get_type(),
            children = tree.extract_children(id_node)
        );
    }
}