Problem:
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Binary Search Tree I

Search trees are data structures that support dynamic set operations including insert, search, delete and so on. Thus a search tree can be used both as a dictionary and as a priority queue.

Binary search tree is one of fundamental search trees. The keys in a binary search tree are always stored in such a way as to satisfy the following binary search tree property:

Let $x$ be a node in a binary search tree. If $y$ is a node in the left subtree of $x$, then $y.key \leq x.key$. If $y$ is a node in the right subtree of $x$, then $x.key \leq y.key$.

The following figure shows an example of the binary search tree.

For example, keys of nodes which belong to the left sub-tree of the node containing 80 are less than or equal to 80, and keys of nodes which belong to the right sub-tree are more than or equal to 80. The binary search tree property allows us to print out all the keys in the tree in sorted order by an inorder tree walk.

A binary search tree should be implemented in such a way that the binary search tree property continues to hold after modifications by insertions and deletions. A binary search tree can be represented by a linked data structure in which each node is an object. In addition to a key field and satellite data, each node contains fields left, right, and p that point to the nodes corresponding to its left child, its right child, and its parent, respectively.

To insert a new value $v$ into a binary search tree $T$, we can use the procedure insert as shown in the following pseudo code. The insert procedure is passed a node $z$ for which $z.key = v$, $z.left = NIL$, and $z.right = NIL$. The procedure modifies $T$ and some of the fields of $z$ in such a way that $z$ is inserted into an appropriate position in the tree.

1 insert(T, z)
2 y = NIL // parent of x
3 x = 'the root of T'
4 while x ≠ NIL
5 y = x // set the parent
6 if z.key < x.key
7 x = x.left // move to the left child
8 else
9 x = x.right // move to the right child
10 z.p = y
11
12 if y == NIL // T is empty
13 'the root of T' = z
14 else if z.key < y.key
15 y.left = z // z is the left child of y
16 else
17 y.right = z // z is the right child of y

Write a program which performs the following operations to a binary search tree $T$.

insert $k$: Insert a node containing $k$ as key into $T$.

print: Print the keys of the binary search tree by inorder tree walk and preorder tree walk respectively.

You should use the above pseudo code to implement the insert operation. $T$ is empty at the initial state.

Input

In the first line, the number of operations $m$ is given. In the following $m$ lines, operations represented by insert $k$ or print are given.

Output

For each print operation, print a list of keys obtained by inorder tree walk and preorder tree walk in a line respectively. Put a space character before each key.

Constraints

The number of operations $\leq 500,000$

The number of print operations $\leq 10$.

$-2,000,000,000 \leq key \leq 2,000,000,000$

The height of the binary tree does not exceed 100 if you employ the above pseudo code.

The keys in the binary search tree are all different.

Sample Input 1

8
insert 30
insert 88
insert 12
insert 1
insert 20
insert 17
insert 25
print

Sample Output 1

1 12 17 20 25 30 88
30 12 1 20 17 25 88

Reference

Introduction to Algorithms, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The MIT Press.

Solution in Python3 (judge verdict: Accepted)


class Node:

    def __init__(self, key: int) -> None:
        self.key = key
        self.parent = None
        self.left = None
        self.right = None


def input_line() -> (str, int):
    line = input().split()
    if line[0] == 'insert':
        return 'insert', int(line[1])
    else:
        return line[0], None


def preorder(tree: Node) -> None:
    """先行順巡回。"""
    print(f' {tree.key}', end='')
    if tree.left is not None:
        preorder(tree.left)
    if tree.right is not None:
        preorder(tree.right)


def inorder(tree: Node) -> None:
    """中間順巡回。"""
    if tree.left is not None:
        inorder(tree.left)
    print(f' {tree.key}', end='')
    if tree.right is not None:
        inorder(tree.right)


def insert(new_node: Node) -> None:
    """二分探索木に挿入する。"""
    global root  # rootはglobalに共有する。
    parent = None
    x = root

    # new_nodeを付け加える節(parent)を探索。
    while x is not None:
        parent = x
        if new_node.key < x.key:
            x = x.left
        else:
            x = x.right
    new_node.parent = parent

    # new_nodeの子供を設定。
    if parent is None:  # treeが空だった場合は、rootにnew_nodeを設定。
        # print(f'key: {new_node.key} op: new node')
        root = new_node
    elif new_node.key < parent.key:
        # print(f'key: {new_node.key} op: left')
        parent.left = new_node
    else:
        # print(f'key: {new_node.key} op: right')
        parent.right = new_node


root = None  # treeの根を表す。初期値はNone。
N = int(input())
for _i in range(N):
    order, val = input_line()
    if order == 'insert':
        node = Node(val)
        insert(node)
    else:
        inorder(root)
        print()
        preorder(root)
        print()