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 OCaml (judge verdict: Accepted)

type tree = Nil | Node of int * tree * tree

let create () = Nil

let insert x t =
  let rec doit = function
    | Nil -> Node (x, Nil, Nil)
    | Node (y, l, r) ->
      if x < y then Node (y, doit l, r)
      else Node (y, l, doit r) in
  doit t

let preorder f t =
  let rec doit = function
    | Nil -> ()
    | Node (x, l, r) -> f x; doit l; doit r in
  doit t

let inorder f t =
  let rec doit = function
    | Nil -> ()
    | Node (x, l, r) -> doit l; f x; doit r in
  doit t

let print t =
  inorder (fun e -> print_string " "; print_int e) t;
  print_newline ();
  preorder (fun e -> print_string " "; print_int e) t;
  print_newline ()

let find x t =
  let rec doit = function
    | Nil -> false
    | Node (y, _, _) when x = y -> true
    | Node (y, l, _) when x < y -> doit l
    | Node (_, _, r) -> doit r in
  doit t

let delete x t =
  let rec balance = function
    | Nil -> assert false
    | Node (x, Nil, r) -> (x, r)
    | Node (x, l, r) ->
      let (m, l) = balance l in
      (m, Node (x, l, r)) in
  let rec doit = function
    | Nil -> assert false
    | Node (y, l, r) ->
      if x = y then
        if l = Nil then r
        else if r = Nil then l
        else
          let (m, r) = balance r in
          Node (m, l, r)
      else if x < y then Node (y, doit l, r)
      else Node (y, l, doit r) in
  doit t

let split_on_char sep s =
  let open String in
  let r = ref [] in
  let j = ref (length s) in
  for i = length s - 1 downto 0 do
    if get s i = sep then begin
      r := sub s (i + 1) (!j - i - 1) :: !r;
      j := i
    end
  done;
  sub s 0 !j :: !r

let () =
  let m = read_int () in
  let t = ref (create ()) in
  for _ = 0 to m - 1 do
    match read_line () |> split_on_char ' ' with
    | ["insert"; n] -> t := insert (int_of_string n) !t
    | ["find"; n] -> print_endline (if find (int_of_string n) !t then "yes" else "no")
    | ["delete"; n] -> t := delete (int_of_string n) !t
    | _ -> print !t
  done