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// C++ program to delete a node from AVL Tree
#include <bits/stdc++.h>
using namespace std;

// An AVL tree node
class Node
{
public:
    int key;
    Node *left;
    Node *right;
    int height;
};

// A utility function to get maximum
// of two integers
int max(int a, int b);

// A utility function to get height
// of the tree
int height(Node *N)
{
    if (N == NULL)
        return 0;
    return N->height;
}

// A utility function to get maximum
// of two integers
int max(int a, int b)
{
    return (a > b) ? a : b;
}

/* Helper function that allocates a
new node with the given key and
NULL left and right pointers. */
Node *newNode(int key)
{
    Node *node = new Node();
    node->key = key;
    node->left = NULL;
    node->right = NULL;
    node->height = 1; // new node is initially
                      // added at leaf
    return (node);
}

// A utility function to right
// rotate subtree rooted with y
// See the diagram given above.
Node *rightRotate(Node *y)
{
    Node *x = y->left;
    Node *T2 = x->right;

    // Perform rotation
    x->right = y;
    y->left = T2;

    // Update heights
    y->height = max(height(y->left),
                    height(y->right)) +
                1;
    x->height = max(height(x->left),
                    height(x->right)) +
                1;

    // Return new root
    return x;
}

// A utility function to left
// rotate subtree rooted with x
// See the diagram given above.
Node *leftRotate(Node *x)
{
    Node *y = x->right;
    Node *T2 = y->left;

    // Perform rotation
    y->left = x;
    x->right = T2;

    // Update heights
    x->height = max(height(x->left),
                    height(x->right)) +
                1;
    y->height = max(height(y->left),
                    height(y->right)) +
                1;

    // Return new root
    return y;
}

// Get Balance factor of node N
int getBalance(Node *N)
{
    if (N == NULL)
        return 0;
    return height(N->left) -
           height(N->right);
}

Node *insert(Node *node, int key)
{
    /* 1. Perform the normal BST rotation */
    if (node == NULL)
        return (newNode(key));

    if (key < node->key)
        node->left = insert(node->left, key);
    else if (key > node->key)
        node->right = insert(node->right, key);
    else // Equal keys not allowed
        return node;

    /* 2. Update height of this ancestor node */
    node->height = 1 + max(height(node->left),
                           height(node->right));

    /* 3. Get the balance factor of this
        ancestor node to check whether
        this node became unbalanced */
    int balance = getBalance(node);

    // If this node becomes unbalanced,
    // then there are 4 cases

    // Left Left Case
    if (balance > 1 && key < node->left->key)
        return rightRotate(node);

    // Right Right Case
    if (balance < -1 && key > node->right->key)
        return leftRotate(node);

    // Left Right Case
    if (balance > 1 && key > node->left->key)
    {
        node->left = leftRotate(node->left);
        return rightRotate(node);
    }

    // Right Left Case
    if (balance < -1 && key < node->right->key)
    {
        node->right = rightRotate(node->right);
        return leftRotate(node);
    }

    /* return the (unchanged) node pointer */
    return node;
}

/* Given a non-empty binary search tree,
return the node with minimum key value
found in that tree. Note that the entire
tree does not need to be searched. */
Node *minValueNode(Node *node)
{
    Node *current = node;

    /* loop down to find the leftmost leaf */
    while (current->left != NULL)
        current = current->left;

    return current;
}

// Recursive function to delete a node
// with given key from subtree with
// given root. It returns root of the
// modified subtree.
Node *deleteNode(Node *root, int key)
{

    // STEP 1: PERFORM STANDARD BST DELETE
    if (root == NULL)
        return root;

    // If the key to be deleted is smaller
    // than the root's key, then it lies
    // in left subtree
    if (key < root->key)
        root->left = deleteNode(root->left, key);

    // If the key to be deleted is greater
    // than the root's key, then it lies
    // in right subtree
    else if (key > root->key)
        root->right = deleteNode(root->right, key);

    // if key is same as root's key, then
    // This is the node to be deleted
    else
    {
        // node with only one child or no child
        if ((root->left == NULL) ||
            (root->right == NULL))
        {
            Node *temp = root->left ? root->left : root->right;

            // No child case
            if (temp == NULL)
            {
                temp = root;
                root = NULL;
            }
            else               // One child case
                *root = *temp; // Copy the contents of
                               // the non-empty child
            free(temp);
        }
        else
        {
            // node with two children: Get the inorder
            // successor (smallest in the right subtree)
            Node *temp = minValueNode(root->right);

            // Copy the inorder successor's
            // data to this node
            root->key = temp->key;

            // Delete the inorder successor
            root->right = deleteNode(root->right,
                                     temp->key);
        }
    }

    // If the tree had only one node
    // then return
    if (root == NULL)
        return root;

    // STEP 2: UPDATE HEIGHT OF THE CURRENT NODE
    root->height = 1 + max(height(root->left),
                           height(root->right));

    // STEP 3: GET THE BALANCE FACTOR OF
    // THIS NODE (to check whether this
    // node became unbalanced)
    int balance = getBalance(root);

    // If this node becomes unbalanced,
    // then there are 4 cases

    // Left Left Case
    if (balance > 1 &&
        getBalance(root->left) >= 0)
        return rightRotate(root);

    // Left Right Case
    if (balance > 1 &&
        getBalance(root->left) < 0)
    {
        root->left = leftRotate(root->left);
        return rightRotate(root);
    }

    // Right Right Case
    if (balance < -1 &&
        getBalance(root->right) <= 0)
        return leftRotate(root);

    // Right Left Case
    if (balance < -1 &&
        getBalance(root->right) > 0)
    {
        root->right = rightRotate(root->right);
        return leftRotate(root);
    }

    return root;
}

// A utility function to print preorder
// traversal of the tree.
// The function also prints height
// of every node
void preOrder(Node *root)
{
    if (root != NULL)
    {
        cout << root->key << " ";
        preOrder(root->left);
        preOrder(root->right);
    }
}

// Driver Code
int main()
{
    Node *root = NULL;

    /* Constructing tree given in
    the above figure */
    root = insert(root, 9);
    root = insert(root, 5);
    root = insert(root, 10);
    root = insert(root, 0);
    root = insert(root, 6);
    root = insert(root, 11);
    root = insert(root, -1);
    root = insert(root, 1);
    root = insert(root, 2);

    /* The constructed AVL Tree would be
            9
        / \
        1 10
        / \ \
    0 5 11
    / / \
    -1 2 6
    */

    cout << "Preorder traversal of the "
            "constructed AVL tree is \n";
    preOrder(root);

    root = deleteNode(root, 10);

    /* The AVL Tree after deletion of 10
            1
        / \
        0 9
        / / \
    -1 5	 11
        / \
        2 6
    */

    cout << "\nPreorder traversal after"
         << " deletion of 10 \n";
    preOrder(root);

    return 0;
}

//-------------------------------
// search b tree

// Searching a key on a B-tree in C++

#include <iostream>
using namespace std;

class TreeNode
{
    int *keys;
    int t;
    TreeNode **C;
    int n;
    bool leaf;

public:
    TreeNode(int temp, bool bool_leaf);

    void insertNonFull(int k);
    void splitChild(int i, TreeNode *y);
    void traverse();

    TreeNode *search(int k);

    friend class BTree;
};

class BTree
{
    TreeNode *root;
    int t;

public:
    BTree(int temp)
    {
        root = NULL;
        t = temp;
    }

    void traverse()
    {
        if (root != NULL)
            root->traverse();
    }

    TreeNode *search(int k)
    {
        return (root == NULL) ? NULL : root->search(k);
    }

    void insert(int k);
};

TreeNode::TreeNode(int t1, bool leaf1)
{
    t = t1;
    leaf = leaf1;

    keys = new int[2 * t - 1];
    C = new TreeNode *[2 * t];

    n = 0;
}

void TreeNode::traverse()
{
    int i;
    for (i = 0; i < n; i++)
    {
        if (leaf == false)
            C[i]->traverse();
        cout << " " << keys[i];
    }

    if (leaf == false)
        C[i]->traverse();
}

TreeNode *TreeNode::search(int k)
{
    int i = 0;
    while (i < n && k > keys[i])
        i++;

    if (keys[i] == k)
        return this;

    if (leaf == true)
        return NULL;

    return C[i]->search(k);
}

void BTree::insert(int k)
{
    if (root == NULL)
    {
        root = new TreeNode(t, true);
        root->keys[0] = k;
        root->n = 1;
    }
    else
    {
        if (root->n == 2 * t - 1)
        {
            TreeNode *s = new TreeNode(t, false);

            s->C[0] = root;

            s->splitChild(0, root);

            int i = 0;
            if (s->keys[0] < k)
                i++;
            s->C[i]->insertNonFull(k);

            root = s;
        }
        else
            root->insertNonFull(k);
    }
}

void TreeNode::insertNonFull(int k)
{
    int i = n - 1;

    if (leaf == true)
    {
        while (i >= 0 && keys[i] > k)
        {
            keys[i + 1] = keys[i];
            i--;
        }

        keys[i + 1] = k;
        n = n + 1;
    }
    else
    {
        while (i >= 0 && keys[i] > k)
            i--;

        if (C[i + 1]->n == 2 * t - 1)
        {
            splitChild(i + 1, C[i + 1]);

            if (keys[i + 1] < k)
                i++;
        }
        C[i + 1]->insertNonFull(k);
    }
}

void TreeNode::splitChild(int i, TreeNode *y)
{
    TreeNode *z = new TreeNode(y->t, y->leaf);
    z->n = t - 1;

    for (int j = 0; j < t - 1; j++)
        z->keys[j] = y->keys[j + t];

    if (y->leaf == false)
    {
        for (int j = 0; j < t; j++)
            z->C[j] = y->C[j + t];
    }

    y->n = t - 1;
    for (int j = n; j >= i + 1; j--)
        C[j + 1] = C[j];

    C[i + 1] = z;

    for (int j = n - 1; j >= i; j--)
        keys[j + 1] = keys[j];

    keys[i] = y->keys[t - 1];
    n = n + 1;
}

int main()
{
    BTree t(3);
    t.insert(8);
    t.insert(9);
    t.insert(10);
    t.insert(11);
    t.insert(15);
    t.insert(16);
    t.insert(17);
    t.insert(18);
    t.insert(20);
    t.insert(23);

    cout << "The B-tree is: ";
    t.traverse();

    int k = 10;
    (t.search(k) != NULL) ? cout << endl
                                 << k << " is found"
                          : cout << endl
                                 << k << " is not Found";

    k = 2;
    (t.search(k) != NULL) ? cout << endl
                                 << k << " is found"
                          : cout << endl
                                 << k << " is not Found\n";
}
//-------------------------------

// Inserting a key on a B-tree in C++

#include <iostream>
using namespace std;

class Node
{
    int *keys;
    int t;
    Node **C;
    int n;
    bool leaf;

public:
    Node(int _t, bool _leaf);

    void insertNonFull(int k);
    void splitChild(int i, Node *y);
    void traverse();

    friend class BTree;
};

class BTree
{
    Node *root;
    int t;

public:
    BTree(int _t)
    {
        root = NULL;
        t = _t;
    }

    void traverse()
    {
        if (root != NULL)
            root->traverse();
    }

    void insert(int k);
};

Node::Node(int t1, bool leaf1)
{
    t = t1;
    leaf = leaf1;

    keys = new int[2 * t - 1];
    C = new Node *[2 * t];

    n = 0;
}

// Traverse the nodes
void Node::traverse()
{
    int i;
    for (i = 0; i < n; i++)
    {
        if (leaf == false)
            C[i]->traverse();
        cout << " " << keys[i];
    }

    if (leaf == false)
        C[i]->traverse();
}

// Insert the node
void BTree::insert(int k)
{
    if (root == NULL)
    {
        root = new Node(t, true);
        root->keys[0] = k;
        root->n = 1;
    }
    else
    {
        if (root->n == 2 * t - 1)
        {
            Node *s = new Node(t, false);

            s->C[0] = root;

            s->splitChild(0, root);

            int i = 0;
            if (s->keys[0] < k)
                i++;
            s->C[i]->insertNonFull(k);

            root = s;
        }
        else
            root->insertNonFull(k);
    }
}

// Insert non full condition
void Node::insertNonFull(int k)
{
    int i = n - 1;

    if (leaf == true)
    {
        while (i >= 0 && keys[i] > k)
        {
            keys[i + 1] = keys[i];
            i--;
        }

        keys[i + 1] = k;
        n = n + 1;
    }
    else
    {
        while (i >= 0 && keys[i] > k)
            i--;

        if (C[i + 1]->n == 2 * t - 1)
        {
            splitChild(i + 1, C[i + 1]);

            if (keys[i + 1] < k)
                i++;
        }
        C[i + 1]->insertNonFull(k);
    }
}

// split the child
void Node::splitChild(int i, Node *y)
{
    Node *z = new Node(y->t, y->leaf);
    z->n = t - 1;

    for (int j = 0; j < t - 1; j++)
        z->keys[j] = y->keys[j + t];

    if (y->leaf == false)
    {
        for (int j = 0; j < t; j++)
            z->C[j] = y->C[j + t];
    }

    y->n = t - 1;
    for (int j = n; j >= i + 1; j--)
        C[j + 1] = C[j];

    C[i + 1] = z;

    for (int j = n - 1; j >= i; j--)
        keys[j + 1] = keys[j];

    keys[i] = y->keys[t - 1];
    n = n + 1;
}

int main()
{
    BTree t(3);
    t.insert(8);
    t.insert(9);
    t.insert(10);
    t.insert(11);
    t.insert(15);
    t.insert(16);
    t.insert(17);
    t.insert(18);
    t.insert(20);
    t.insert(23);

    cout << "The B-tree is: ";
    t.traverse();
}
//-------------------------------

// Deleting a key from a B-tree in C++

#include <iostream>
using namespace std;

class BTreeNode
{
    int *keys;
    int t;
    BTreeNode **C;
    int n;
    bool leaf;

public:
    BTreeNode(int _t, bool _leaf);

    void traverse();

    int findKey(int k);
    void insertNonFull(int k);
    void splitChild(int i, BTreeNode *y);
    void deletion(int k);
    void removeFromLeaf(int idx);
    void removeFromNonLeaf(int idx);
    int getPredecessor(int idx);
    int getSuccessor(int idx);
    void fill(int idx);
    void borrowFromPrev(int idx);
    void borrowFromNext(int idx);
    void merge(int idx);
    friend class BTree;
};

class BTree
{
    BTreeNode *root;
    int t;

public:
    BTree(int _t)
    {
        root = NULL;
        t = _t;
    }

    void traverse()
    {
        if (root != NULL)
            root->traverse();
    }

    void insertion(int k);

    void deletion(int k);
};

// B tree node
BTreeNode::BTreeNode(int t1, bool leaf1)
{
    t = t1;
    leaf = leaf1;

    keys = new int[2 * t - 1];
    C = new BTreeNode *[2 * t];

    n = 0;
}

// Find the key
int BTreeNode::findKey(int k)
{
    int idx = 0;
    while (idx < n && keys[idx] < k)
        ++idx;
    return idx;
}

// Deletion operation
void BTreeNode::deletion(int k)
{
    int idx = findKey(k);

    if (idx < n && keys[idx] == k)
    {
        if (leaf)
            removeFromLeaf(idx);
        else
            removeFromNonLeaf(idx);
    }
    else
    {
        if (leaf)
        {
            cout << "The key " << k << " is does not exist in the tree\n";
            return;
        }

        bool flag = ((idx == n) ? true : false);

        if (C[idx]->n < t)
            fill(idx);

        if (flag && idx > n)
            C[idx - 1]->deletion(k);
        else
            C[idx]->deletion(k);
    }
    return;
}

// Remove from the leaf
void BTreeNode::removeFromLeaf(int idx)
{
    for (int i = idx + 1; i < n; ++i)
        keys[i - 1] = keys[i];

    n--;

    return;
}

// Delete from non leaf node
void BTreeNode::removeFromNonLeaf(int idx)
{
    int k = keys[idx];

    if (C[idx]->n >= t)
    {
        int pred = getPredecessor(idx);
        keys[idx] = pred;
        C[idx]->deletion(pred);
    }

    else if (C[idx + 1]->n >= t)
    {
        int succ = getSuccessor(idx);
        keys[idx] = succ;
        C[idx + 1]->deletion(succ);
    }

    else
    {
        merge(idx);
        C[idx]->deletion(k);
    }
    return;
}

int BTreeNode::getPredecessor(int idx)
{
    BTreeNode *cur = C[idx];
    while (!cur->leaf)
        cur = cur->C[cur->n];

    return cur->keys[cur->n - 1];
}

int BTreeNode::getSuccessor(int idx)
{
    BTreeNode *cur = C[idx + 1];
    while (!cur->leaf)
        cur = cur->C[0];

    return cur->keys[0];
}

void BTreeNode::fill(int idx)
{
    if (idx != 0 && C[idx - 1]->n >= t)
        borrowFromPrev(idx);

    else if (idx != n && C[idx + 1]->n >= t)
        borrowFromNext(idx);

    else
    {
        if (idx != n)
            merge(idx);
        else
            merge(idx - 1);
    }
    return;
}

// Borrow from previous
void BTreeNode::borrowFromPrev(int idx)
{
    BTreeNode *child = C[idx];
    BTreeNode *sibling = C[idx - 1];

    for (int i = child->n - 1; i >= 0; --i)
        child->keys[i + 1] = child->keys[i];

    if (!child->leaf)
    {
        for (int i = child->n; i >= 0; --i)
            child->C[i + 1] = child->C[i];
    }

    child->keys[0] = keys[idx - 1];

    if (!child->leaf)
        child->C[0] = sibling->C[sibling->n];

    keys[idx - 1] = sibling->keys[sibling->n - 1];

    child->n += 1;
    sibling->n -= 1;

    return;
}

// Borrow from the next
void BTreeNode::borrowFromNext(int idx)
{
    BTreeNode *child = C[idx];
    BTreeNode *sibling = C[idx + 1];

    child->keys[(child->n)] = keys[idx];

    if (!(child->leaf))
        child->C[(child->n) + 1] = sibling->C[0];

    keys[idx] = sibling->keys[0];

    for (int i = 1; i < sibling->n; ++i)
        sibling->keys[i - 1] = sibling->keys[i];

    if (!sibling->leaf)
    {
        for (int i = 1; i <= sibling->n; ++i)
            sibling->C[i - 1] = sibling->C[i];
    }

    child->n += 1;
    sibling->n -= 1;

    return;
}

// Merge
void BTreeNode::merge(int idx)
{
    BTreeNode *child = C[idx];
    BTreeNode *sibling = C[idx + 1];

    child->keys[t - 1] = keys[idx];

    for (int i = 0; i < sibling->n; ++i)
        child->keys[i + t] = sibling->keys[i];

    if (!child->leaf)
    {
        for (int i = 0; i <= sibling->n; ++i)
            child->C[i + t] = sibling->C[i];
    }

    for (int i = idx + 1; i < n; ++i)
        keys[i - 1] = keys[i];

    for (int i = idx + 2; i <= n; ++i)
        C[i - 1] = C[i];

    child->n += sibling->n + 1;
    n--;

    delete (sibling);
    return;
}

// Insertion operation
void BTree::insertion(int k)
{
    if (root == NULL)
    {
        root = new BTreeNode(t, true);
        root->keys[0] = k;
        root->n = 1;
    }
    else
    {
        if (root->n == 2 * t - 1)
        {
            BTreeNode *s = new BTreeNode(t, false);

            s->C[0] = root;

            s->splitChild(0, root);

            int i = 0;
            if (s->keys[0] < k)
                i++;
            s->C[i]->insertNonFull(k);

            root = s;
        }
        else
            root->insertNonFull(k);
    }
}

// Insertion non full
void BTreeNode::insertNonFull(int k)
{
    int i = n - 1;

    if (leaf == true)
    {
        while (i >= 0 && keys[i] > k)
        {
            keys[i + 1] = keys[i];
            i--;
        }

        keys[i + 1] = k;
        n = n + 1;
    }
    else
    {
        while (i >= 0 && keys[i] > k)
            i--;

        if (C[i + 1]->n == 2 * t - 1)
        {
            splitChild(i + 1, C[i + 1]);

            if (keys[i + 1] < k)
                i++;
        }
        C[i + 1]->insertNonFull(k);
    }
}

// Split child
void BTreeNode::splitChild(int i, BTreeNode *y)
{
    BTreeNode *z = new BTreeNode(y->t, y->leaf);
    z->n = t - 1;

    for (int j = 0; j < t - 1; j++)
        z->keys[j] = y->keys[j + t];

    if (y->leaf == false)
    {
        for (int j = 0; j < t; j++)
            z->C[j] = y->C[j + t];
    }

    y->n = t - 1;

    for (int j = n; j >= i + 1; j--)
        C[j + 1] = C[j];

    C[i + 1] = z;

    for (int j = n - 1; j >= i; j--)
        keys[j + 1] = keys[j];

    keys[i] = y->keys[t - 1];

    n = n + 1;
}

// Traverse
void BTreeNode::traverse()
{
    int i;
    for (i = 0; i < n; i++)
    {
        if (leaf == false)
            C[i]->traverse();
        cout << " " << keys[i];
    }

    if (leaf == false)
        C[i]->traverse();
}

// Delete Operation
void BTree::deletion(int k)
{
    if (!root)
    {
        cout << "The tree is empty\n";
        return;
    }

    root->deletion(k);

    if (root->n == 0)
    {
        BTreeNode *tmp = root;
        if (root->leaf)
            root = NULL;
        else
            root = root->C[0];

        delete tmp;
    }
    return;
}

int main()
{
    BTree t(3);
    t.insertion(8);
    t.insertion(9);
    t.insertion(10);
    t.insertion(11);
    t.insertion(15);
    t.insertion(16);
    t.insertion(17);
    t.insertion(18);
    t.insertion(20);
    t.insertion(23);

    cout << "The B-tree is: ";
    t.traverse();

    t.deletion(20);

    cout << "\nThe B-tree is: ";
    t.traverse();
}