Problem:
Binary Search Tree III

Write a program which performs the following operations to a binary search tree $T$ by adding delete operation to B: Binary Search Tree II.

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

find $k$: Report whether $T$ has a node containing $k$.

delete $k$: Delete a node containing $k$.

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

The operation delete $k$ for deleting a given node $z$ containing key $k$ from $T$ can be implemented by an algorithm which considers the following cases:

If $z$ has no children, we modify its parent $z.p$ to replace $z$ with NIL as its child (delete $z$).

If $z$ has only a single child, we "splice out" $z$ by making a new link between its child and its parent.

If $z$ has two children, we splice out $z$'s successor $y$ and replace $z$'s key with $y$'s key.

Input

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

Output

For each find $k$ operation, print "yes" if $T$ has a node containing $k$, "no" if not.

In addition, 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

18
insert 8
insert 2
insert 3
insert 7
insert 22
insert 1
find 1
find 2
find 3
find 4
find 5
find 6
find 7
find 8
print
delete 3
delete 7
print

Sample Output 1

yes
yes
yes
no
no
no
yes
yes
1 2 3 7 8 22
8 2 1 3 7 22
1 2 8 22
8 2 1 22

Reference

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

Solution in Ruby (judge verdict: Accepted)

class BinarySearchTree
	class Node
		attr_reader :key
		attr_accessor :parent, :left, :right
		def initialize(key, parent)
			@key = key
			@parent = parent
		end
	end

	attr_accessor :root

	def insert(key)
		parent  = nil
		current = @root
		while current
			parent = current
			if key < current.key
				current = current.left
			else
				current = current.right
			end
		end

		new_node = Node.new(key, parent)
		if !parent
			@root = new_node
		elsif key < parent.key
			parent.left = new_node
		else
			parent.right = new_node
		end
		self
	end

	def find(key)
		current = @root
		while current && current.key != key
			if key < current.key
				current = current.left
			else
				current = current.right
			end
		end
		current
	end

	def delete(key)
		target = find(key)
		return nil unless target
		delete_helper(target)
	end

	def keys_inorder
		inorder_helper(@root)
	end

	def keys_preorder
		preorder_helper(@root)
	end

	private
	def copy_node(node, new_key=nil)
		new_key ||= node.key

		copied = Node.new(new_key, node.parent)
		copied.left  = node.left
		copied.right = node.right
		copied
	end

	def replace(node, new_node)
		parent = node.parent
		new_node.parent = parent if new_node

		if !parent
			@root = new_node
		elsif parent.left && parent.left.key == node.key
			parent.left = new_node
		else
			parent.right = new_node
		end

		node
	end

	def delete_helper(node)
		if node.left && node.right
			next_node_index = inorder_helper(node.right).first
			next_node = find(next_node_index)

			copied = copy_node(node, next_node.key)
			node.right.parent = copied
			node.left.parent  = copied
			replace(node, copied)

			delete_helper(next_node)
			return node # 元々の削除対象を返したい為
		end

		return replace(node, node.left)  if node.left
		return replace(node, node.right) if node.right
		replace(node, nil)
	end

	def inorder_helper(node)
		return [] unless node
		result = []
		result.concat(inorder_helper(node.left))
		result << node.key
		result.concat(inorder_helper(node.right))
	end

	def preorder_helper(node)
		return [] unless node
		result = [node.key]
		result.concat(preorder_helper(node.left))
		result.concat(preorder_helper(node.right))
	end

end

tree = BinarySearchTree.new
n = STDIN.gets.to_i
n.times do
	command, param = STDIN.gets.split[0..1]
	if command == "insert"
		tree.insert(param.to_i)

	elsif command == "find"
		puts (tree.find(param.to_i) ? "yes" : "no")

	elsif command == "delete"
		tree.delete(param.to_i)

	elsif command == "print"
		puts " #{tree.keys_inorder.join(" ")}"
		puts " #{tree.keys_preorder.join(" ")}"

	else
		raise "Unexpected input #{command}, #{param}"
	end
end