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 MyBinaryNode
  attr_accessor :parent, :left, :right, :v
  attr_reader :id

  def initialize(id, v)
    @id = id
    @parent = nil
    @left = nil
    @right = nil
    @v = v
  end

  def to_s
    @v
  end
end

def pre_order(node)
  if node.nil?
    return 
  end
  print ' '
  print node.v
  pre_order(node.left)
  pre_order(node.right)
end

def in_order(node)
  if node.nil?
    return 
  end
  in_order(node.left)
  print ' '
  print node.v
  in_order(node.right)
end

# def post_order(nodes, id)
#   if id.nil?
#     return 
#   end
#   post_order(nodes, nodes[id].left)
#   post_order(nodes, nodes[id].right)
#   print ' '
#   print id
# end

def print_nodes(node)
  in_order(node)
  puts ""
  pre_order(node)
  puts ""
end

def insert(z, root)
  y = nil
  x = root
  while !x.nil? do
    y = x
    if z.v < x.v
      x = x.left
    else
      x = x.right
    end
  end
  
  z.parent = y

  if y.nil?
    # skip
  elsif z.v < y.v
    y.left = z
  else
    y.right = z
  end

end

def find(x, k)
  while !x.nil? && k != x.v do
    if k < x.v
      x = x.left
    else
      x = x.right
    end
  end

  x
end

def delete(z)
  if z.left.nil? && z.right.nil?
    p = z.parent
    if p.left&.v == z.v
      p.left = nil
    else
      p.right = nil
    end
  elsif z.left && z.right
    next_node = get_successor(z)
    z.v = next_node.v
    delete(next_node)
  else
    p = z.parent
    c = z.left || z.right
    if p.left&.v == z.v
      p.left = c
    else
      p.right = c
    end
    c.parent = p
    
  end
end

def get_successor(x)
  if x.right
    return get_minimum(x.right)
  end

  y = x.parent
  while y && x == y.right
    x = y
    y = y.parent
  end
  return y
end

def get_minimum(x)
  while x.left
    x = x.left
  end
  x
end


n = gets.to_i
root = nil
i = 0
n.times do
  order, v = gets.split
  v = v.to_i
  case order
  when "insert"
    node = MyBinaryNode.new(i, v)
    insert(node, root)
    if i == 0
      root = node
    end
    i += 1
  when "print"
    print_nodes(root)
  when "find"
    if find(root, v)
      puts "yes"
    else
      puts "no"
    end
  when "delete"
    target_node = find(root, v)
    delete(target_node)
  end
end