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)

m = gets.chomp.to_i
commands = []
m.times do 
    cmd, v = gets.chomp.split(' ')
    v = v.to_i unless v.nil?
    commands.push([cmd, v])
end

def insert(tree, z)
    y = nil
    x = tree[:root]
    # p x, y, z
    until x.nil? do
        y = x
        # puts "x[:key] = #{x[:key]}"
        # puts "z[:key] = #{z[:key]}"
        if z[:key] < x[:key] then
            x = tree[x[:l]]
        else
            x = tree[x[:r]]
        end
    end
    z[:p] = y[:key] unless y.nil?
    if y.nil? then
        tree[:root] = z
        tree[z[:key]] = z
    elsif z[:key] < y[:key]
        y[:l] = z[:key]
        tree[z[:key]] = z
    else
        y[:r] = z[:key]
        tree[z[:key]] = z
    end
end

def getMinimium(tree, z)
    return z if tree[z][:l] == nil
    min = getMinimium(tree, tree[z][:l])
    return min
end

def getSuccessor(tree, z)
    s = getMinimium(tree, tree[z][:r])
    return s
end

def delete(tree, z)
    return unless tree.has_key?(z)
    if tree[z][:l] == nil && tree[z][:r] == nil then
        [:l, :r].each do |child|
            tree[tree[z][:p]][child] = nil if tree[tree[z][:p]][child] == z
        end
    elsif tree[z][:l] && tree[z][:r] then
        y = getSuccessor(tree, z)
        # 次節点yをzにコピーしてyは削除する。削除の際は以下略。
        zLOrR = tree[tree[z][:p]][:l] == z ? :l : :r
        tree[tree[z][:p]][zLOrR] = y
        tree[tree[z][:l]][:p] = y
        if tree[y][:p] == z then
            tree[y][:p] = tree[z][:p]
            tree[y][:l] = tree[z][:l]
        else
            tree[tree[z][:r]][:p] = y
            # yLOrR = tree[tree[y][:p]][:l] == y ? :l : :r
            # tree[tree[y][:p]][yLOrR] = nil
            tree[tree[y][:p]][:l] = nil
            tree[y] = tree[z]
            tree[y][:key] = y
        end
    else
        onlyChild = tree[z][:l].nil? ? :r : :l
        # zが親のlかrかを特定、そこをzの子（lかrか）にすげかえる
        [:l, :r].each do |child|
            tree[tree[z][:p]][child] = tree[z][onlyChild] if tree[tree[z][:p]][child] == z
        end
        # zの子（lかrか）の親をzの親にすげかえる
        tree[tree[z][onlyChild]][:p] = tree[z][:p]
    end
    tree.delete(z)
end

def preParse(tree, key)
    return if key == nil
    print " #{key}"
    preParse(tree, tree[key][:l])
    preParse(tree, tree[key][:r])    
end

def inParse(tree, key)
    return if key == nil
    inParse(tree, tree[key][:l])
    print " #{key}"
    inParse(tree, tree[key][:r])
end

tree = { root: nil }
commands.each do |cmd|
    if cmd[0] == "insert" then
        node = { key: cmd[1], p: nil, l: nil, r: nil }
        insert(tree, node)
    elsif cmd[0] == "delete" then
        delete(tree, cmd[1])
    elsif cmd[0] == "find" then
        puts tree.has_key?(cmd[1]) ? "yes" : "no" # ずるいけどｗ
    else
        # p tree
        inParse(tree, tree[:root][:key])
        puts "\n"
        preParse(tree, tree[:root][:key])
        puts "\n"
    end
end