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 C (judge verdict: Accepted)

/* Binary Search Tree <Delete> */
#include <stdio.h>
#include <stdlib.h>

struct node
{
    struct node *par;
    struct node *left;
    struct node *right;
    int key;
};

typedef struct node * NodePointer;

void init(void);
NodePointer makenode(int);
void insert(int, NodePointer);
void preorder(NodePointer);
void inorder(NodePointer);
NodePointer find(int, NodePointer);
void delete_node(NodePointer);
NodePointer tree_successor(NodePointer);
NodePointer tree_minimum(NodePointer);

NodePointer head, tail;

int main()
{
    char str[7];
    int i, n, k;
    NodePointer x;

    init();
    scanf("%d", &n);
    for(i=0; i<n; i++){
        scanf("%s", str);
        if(str[0] == 'i'){
            scanf("%d", &k);
            insert(k, head);
        } else if(str[0] == 'p'){
            inorder(head -> right);
            puts("");
            preorder(head -> right);
            puts("");
        } else if(str[0] == 'f'){
            scanf("%d", &k);
            if(find(k, head -> right) != NULL){
                printf("yes\n");
            } else{
                printf("no\n");
            }
        } else{
            scanf("%d", &k);
            x = find(k, head -> right);
            delete_node(x);
        }
    }
    return 0;
}

void init()
{
    head = makenode(-1);
    tail = makenode(-1);
    head -> par = head;
    head -> right = tail;
    head -> left = tail;
}

NodePointer makenode(int k)
{
    NodePointer x;

    x = malloc(sizeof(struct node));
    x -> key = k;
    x -> left = x -> right = tail;

    return x;
}

void insert(int k, NodePointer p)
{
    if(k < p -> key){
        if(p -> left == tail){
            p -> left = makenode(k);
            p -> left -> par = p;
        } else{
            insert(k, p -> left);
        }
    } else{
        if(p -> right == tail){
            p -> right = makenode(k);
            p -> right -> par = p;
        } else{
            insert(k, p -> right);
        }
    }
}

void preorder(NodePointer node)
{
    if(node != tail){
        printf(" %d", node -> key);
        preorder(node -> left);
        preorder(node -> right);
    }
}

void inorder(NodePointer node)
{
    if(node != tail){
        inorder(node -> left);
        printf(" %d", node -> key);
        inorder(node -> right);
    }
}

NodePointer find(int k, NodePointer node)
{
    if(node == tail){
        return NULL;
    }

    if(node -> key == k){
        return node;
    } else if(node -> key < k){
        return find(k, node -> right);
    } else{
        return find(k, node -> left);
    }
}

void delete_node(NodePointer node)
{
    NodePointer x, y;
    if(node -> left == tail || node -> right == tail){
        y = node;
    } else{
        y = tree_successor(node);
    }

    if(node -> left != tail){
        x = y -> left;
    } else{
        x = y -> right;
    }

    if(x != tail){
        x -> par = y -> par;
    }
    if(y -> par == tail){
        node -> par = x;
    } else if(y == y -> par -> left){
        y -> par -> left = x;
    } else{
        y -> par -> right = x;
    }

    if(y != node){
        node -> key = y -> key;
    }
}

NodePointer tree_successor(NodePointer x)
{
    NodePointer y;

    if(x -> right != tail){
        return tree_minimum(x -> right);
    }

    y = x -> par;
    while(y != tail && x == y -> right){
        x = y;
        y = y -> par;
    }
    return y;
}

NodePointer tree_minimum(NodePointer x)
{
    while(x -> left != tail){
        x = x -> left;
    }
    return x;
}