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AVL Tree

AVL Tree
AVL Tree

In this tutorial, you will learn what an AVL Tree is. Additionally, you will discover working instances of different operations performed on an avl tree in C, C++, Java, and Python.

What is an AVL tree?

AVL, named after innovators Adelson-Velsky and Landis, is a binary tree that​ self-balances by keeping a beware of the balance factor of each node.

AVL tree is a self-balancing binary search tree in which every node keeps up additional data called a balance factor whose value is either – 1, 0 or +1.

AVL tree got its name after its inventor Georgy Adelson-Velsky and Landis.


Balance Factor

The balance factor of a node in an AVL tree is the difference between the height of the left subtree and that of the right subtree of that node.

Balance Factor = (Height of Left Subtree – Height of Right Subtree) or (Height of Right Subtree – Height of Left Subtree)

Oneself balancing property of an avl tree is kept up by the balance factor. The value of the balance factor ought to consistently be – 1, 0 or +1.

An example of a balanced avl tree is:


Operations on an AVL tree

Different tasks that can be performed on an AVL tree are:

Rotating the subtrees in an AVL Tree

In rotation activity, the places of the nodes of a subtree are interchanged.


There are two types of rotations:

Left Rotate

In left-rotation, the plan of the nodes on the right is changed into the arrangements on the left node.

Algorithm

  1. Let the initial tree be:

2. If y has a left subtree, assign x as the parent of the left subtree of y.

3. On the off chance that the parent of x is NULL, make y as the root of the tree.

4. Else if x is the left child of p, make y as the left child of p.

5. Else assign y as the right child of p.

6. Make y as the parent of x.


Right Rotate

In left-rotation, the plan of the nodes on the left is changed into the arrangements on the right node.

  1. Let the initial tree be:

2. If x has a right subtree, assign y as the parent of the right subtree of x.

3. On the off chance that the parent of y is NULL, make x as the root of the tree.

4. Else if y is the right child of its parent p, make x as the right child of p.

5. Else assign x as the left child of p.

6. Make x as the parent of y.


Left-Right and Right-Left Rotate

In the left-right rotation, the arrangements are first moved to the left and afterward to the right.

  1. Do left rotation on x-y.

2. Do right rotation on y-z.

In the right-left rotation, the arrangements are first moved to the right and afterward to the left.

  1. Do right rotation on x-y.

2. Do left rotation on z-y.


Algorithm to insert a newnode

A newNode is constantly inserted as a leaf node with a balance factor equivalent to 0.

  1. Let the initial tree be:

Let the node to be inserted be:

2. Go to the fitting leaf node to insert a newNode using the accompanying recursive steps. Compare newKey and rootKey of the current tree.

a. On the off chance that newKey < rootKey, call the addition algorithm on the left subtree of the current node until the leaf node is reached.

b. Else if newKey > rootKey, call addition algorithm on the right subtree of the current node until the leaf node is reached.

c. Else, return leafNode.

3. Compare leafKey acquired from the above steps with newKey:

a. In the event that newKey < leafKey, make newNode as the leftChild of leafNode.

b. Else, make newNode as rightChild of leafNode.

4. Update balanceFactor of the nodes.

5. On the off chance that the nodes are unequal, rebalance the node.

a. On the off chance that balanceFactor > 1, it implies the height of the left subtree is greater than that of the right subtree. Thus, do a right rotation or left-right rotation

a. In the event that newNodeKey < leftChildKey do right rotation.

b. Else, do left-right rotation.

b. In the event that balanceFactor < – 1, it implies the height of the right subtree is greater than that of the left subtree. In this way, do right rotation or right-left rotation

a. In the event that newNodeKey > rightChildKey do left rotation.

b. Else, do right-left rotation

6. The final tree is:


Algorithm to Delete a node

A node is constantly erased as a leaf node. In the wake of erasing a node, the balance factors of the nodes get changed. To rebalance the balance factor, reasonable rotations are performed.

  1. Find nodeToBeDeleted (recursion is used to discover nodeToBeDeleted in the code used beneath).

2. There are three cases for erasing a node:

a. On the off chance that nodeToBeDeleted is the leaf node (ie. doesn’t have any child), at that point eliminate nodeToBeDeleted.

b. In the event that nodeToBeDeleted has one child, substitute the substance of nodeToBeDeleted with that of the child. Eliminate the child.

c. In the event that nodeToBeDeleted has two children, discover the inorder successor w of nodeToBeDeleted (ie. node with a minimum value of key in the right subtree).

a. Substitute the contents of nodeToBeDeleted with that of w.

b. Remove the leaf node w.

3. Update balanceFactor of the nodes.

4. Rebalance the tree if the balance factor of any of the nodes isn’t equivalent to – 1, 0, or 1.

a. On the off chance that balanceFactor of currentNode > 1,

a. In the event that balanceFactor of leftChild >= 0, do right rotation.

b. Else do left-right rotation.

b. On the off chance that balanceFactor of currentNode < – 1,

a. In the event that balanceFactor of rightChild <= 0, do left rotation.

b. Else do right-left rotation.

5. The last tree is:


Python, Java and C/C++ Examples

Python

# AVL tree implementation in Python


import sys

# Create a tree node
class TreeNode(object):
    def __init__(self, key):
        self.key = key
        self.left = None
        self.right = None
        self.height = 1


class AVLTree(object):

    # Function to insert a node
    def insert_node(self, root, key):

        # Find the correct location and insert the node
        if not root:
            return TreeNode(key)
        elif key < root.key:
            root.left = self.insert_node(root.left, key)
        else:
            root.right = self.insert_node(root.right, key)

        root.height = 1 + max(self.getHeight(root.left),
                              self.getHeight(root.right))

        # Update the balance factor and balance the tree
        balanceFactor = self.getBalance(root)
        if balanceFactor > 1:
            if key < root.left.key:
                return self.rightRotate(root)
            else:
                root.left = self.leftRotate(root.left)
                return self.rightRotate(root)

        if balanceFactor < -1:
            if key > root.right.key:
                return self.leftRotate(root)
            else:
                root.right = self.rightRotate(root.right)
                return self.leftRotate(root)

        return root

    # Function to delete a node
    def delete_node(self, root, key):

        # Find the node to be deleted and remove it
        if not root:
            return root
        elif key < root.key:
            root.left = self.delete_node(root.left, key)
        elif key > root.key:
            root.right = self.delete_node(root.right, key)
        else:
            if root.left is None:
                temp = root.right
                root = None
                return temp
            elif root.right is None:
                temp = root.left
                root = None
                return temp
            temp = self.getMinValueNode(root.right)
            root.key = temp.key
            root.right = self.delete_node(root.right,
                                          temp.key)
        if root is None:
            return root

        # Update the balance factor of nodes
        root.height = 1 + max(self.getHeight(root.left),
                              self.getHeight(root.right))

        balanceFactor = self.getBalance(root)

        # Balance the tree
        if balanceFactor > 1:
            if self.getBalance(root.left) >= 0:
                return self.rightRotate(root)
            else:
                root.left = self.leftRotate(root.left)
                return self.rightRotate(root)
        if balanceFactor < -1:
            if self.getBalance(root.right) <= 0:
                return self.leftRotate(root)
            else:
                root.right = self.rightRotate(root.right)
                return self.leftRotate(root)
        return root

    # Function to perform left rotation
    def leftRotate(self, z):
        y = z.right
        T2 = y.left
        y.left = z
        z.right = T2
        z.height = 1 + max(self.getHeight(z.left),
                           self.getHeight(z.right))
        y.height = 1 + max(self.getHeight(y.left),
                           self.getHeight(y.right))
        return y

    # Function to perform right rotation
    def rightRotate(self, z):
        y = z.left
        T3 = y.right
        y.right = z
        z.left = T3
        z.height = 1 + max(self.getHeight(z.left),
                           self.getHeight(z.right))
        y.height = 1 + max(self.getHeight(y.left),
                           self.getHeight(y.right))
        return y

    # Get the height of the node
    def getHeight(self, root):
        if not root:
            return 0
        return root.height

    # Get balance factore of the node
    def getBalance(self, root):
        if not root:
            return 0
        return self.getHeight(root.left) - self.getHeight(root.right)

    def getMinValueNode(self, root):
        if root is None or root.left is None:
            return root
        return self.getMinValueNode(root.left)

    def preOrder(self, root):
        if not root:
            return
        print("{0} ".format(root.key), end="")
        self.preOrder(root.left)
        self.preOrder(root.right)

    # Print the tree
    def printHelper(self, currPtr, indent, last):
        if currPtr != None:
            sys.stdout.write(indent)
            if last:
                sys.stdout.write("R----")
                indent += "     "
            else:
                sys.stdout.write("L----")
                indent += "|    "
            print(currPtr.key)
            self.printHelper(currPtr.left, indent, False)
            self.printHelper(currPtr.right, indent, True)


myTree = AVLTree()
root = None
nums = [33, 13, 52, 9, 21, 61, 8, 11]
for num in nums:
    root = myTree.insert_node(root, num)
myTree.printHelper(root, "", True)
key = 13
root = myTree.delete_node(root, key)
print("After Deletion: ")
myTree.printHelper(root, "", True)

Java

// AVL tree implementation in Java

// Create node
class Node {
  int item, height;
  Node left, right;

  Node(int d) {
    item = d;
    height = 1;
  }
}

// Tree class
class AVLTree {
  Node root;

  int height(Node N) {
    if (N == null)
      return 0;
    return N.height;
  }

  int max(int a, int b) {
    return (a > b) ? a : b;
  }

  Node rightRotate(Node y) {
    Node x = y.left;
    Node T2 = x.right;
    x.right = y;
    y.left = T2;
    y.height = max(height(y.left), height(y.right)) + 1;
    x.height = max(height(x.left), height(x.right)) + 1;
    return x;
  }

  Node leftRotate(Node x) {
    Node y = x.right;
    Node T2 = y.left;
    y.left = x;
    x.right = T2;
    x.height = max(height(x.left), height(x.right)) + 1;
    y.height = max(height(y.left), height(y.right)) + 1;
    return y;
  }

  // Get balance factor of a node
  int getBalanceFactor(Node N) {
    if (N == null)
      return 0;
    return height(N.left) - height(N.right);
  }

  // Insert a node
  Node insertNode(Node node, int item) {

    // Find the position and insert the node
    if (node == null)
      return (new Node(item));
    if (item < node.item)
      node.left = insertNode(node.left, item);
    else if (item > node.item)
      node.right = insertNode(node.right, item);
    else
      return node;

    // Update the balance factor of each node
    // And, balance the tree
    node.height = 1 + max(height(node.left), height(node.right));
    int balanceFactor = getBalanceFactor(node);
    if (balanceFactor > 1) {
      if (item < node.left.item) {
        return rightRotate(node);
      } else if (item > node.left.item) {
        node.left = leftRotate(node.left);
        return rightRotate(node);
      }
    }
    if (balanceFactor < -1) {
      if (item > node.right.item) {
        return leftRotate(node);
      } else if (item < node.right.item) {
        node.right = rightRotate(node.right);
        return leftRotate(node);
      }
    }
    return node;
  }

  Node nodeWithMimumValue(Node node) {
    Node current = node;
    while (current.left != null)
      current = current.left;
    return current;
  }

  // Delete a node
  Node deleteNode(Node root, int item) {

    // Find the node to be deleted and remove it
    if (root == null)
      return root;
    if (item < root.item)
      root.left = deleteNode(root.left, item);
    else if (item > root.item)
      root.right = deleteNode(root.right, item);
    else {
      if ((root.left == null) || (root.right == null)) {
        Node temp = null;
        if (temp == root.left)
          temp = root.right;
        else
          temp = root.left;
        if (temp == null) {
          temp = root;
          root = null;
        } else
          root = temp;
      } else {
        Node temp = nodeWithMimumValue(root.right);
        root.item = temp.item;
        root.right = deleteNode(root.right, temp.item);
      }
    }
    if (root == null)
      return root;

    // Update the balance factor of each node and balance the tree
    root.height = max(height(root.left), height(root.right)) + 1;
    int balanceFactor = getBalanceFactor(root);
    if (balanceFactor > 1) {
      if (getBalanceFactor(root.left) >= 0) {
        return rightRotate(root);
      } else {
        root.left = leftRotate(root.left);
        return rightRotate(root);
      }
    }
    if (balanceFactor < -1) {
      if (getBalanceFactor(root.right) <= 0) {
        return leftRotate(root);
      } else {
        root.right = rightRotate(root.right);
        return leftRotate(root);
      }
    }
    return root;
  }

  void preOrder(Node node) {
    if (node != null) {
      System.out.print(node.item + " ");
      preOrder(node.left);
      preOrder(node.right);
    }
  }

  // Print the tree
  private void printTree(Node currPtr, String indent, boolean last) {
    if (currPtr != null) {
      System.out.print(indent);
      if (last) {
        System.out.print("R----");
        indent += "   ";
      } else {
        System.out.print("L----");
        indent += "|  ";
      }
      System.out.println(currPtr.item);
      printTree(currPtr.left, indent, false);
      printTree(currPtr.right, indent, true);
    }
  }

  // Driver code
  public static void main(String[] args) {
    AVLTree tree = new AVLTree();
    tree.root = tree.insertNode(tree.root, 33);
    tree.root = tree.insertNode(tree.root, 13);
    tree.root = tree.insertNode(tree.root, 53);
    tree.root = tree.insertNode(tree.root, 9);
    tree.root = tree.insertNode(tree.root, 21);
    tree.root = tree.insertNode(tree.root, 61);
    tree.root = tree.insertNode(tree.root, 8);
    tree.root = tree.insertNode(tree.root, 11);
    tree.printTree(tree.root, "", true);
    tree.root = tree.deleteNode(tree.root, 13);
    System.out.println("After Deletion: ");
    tree.printTree(tree.root, "", true);
  }
}

C

// AVL tree implementation in C

#include <stdio.h>
#include <stdlib.h>

// Create Node
struct Node {
  int key;
  struct Node *left;
  struct Node *right;
  int height;
};

int max(int a, int b);

// Calculate height
int height(struct Node *N) {
  if (N == NULL)
    return 0;
  return N->height;
}

int max(int a, int b) {
  return (a > b) ? a : b;
}

// Create a node
struct Node *newNode(int key) {
  struct Node *node = (struct Node *)
    malloc(sizeof(struct Node));
  node->key = key;
  node->left = NULL;
  node->right = NULL;
  node->height = 1;
  return (node);
}

// Right rotate
struct Node *rightRotate(struct Node *y) {
  struct Node *x = y->left;
  struct Node *T2 = x->right;

  x->right = y;
  y->left = T2;

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

  return x;
}

// Left rotate
struct Node *leftRotate(struct Node *x) {
  struct Node *y = x->right;
  struct Node *T2 = y->left;

  y->left = x;
  x->right = T2;

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

  return y;
}

// Get the balance factor
int getBalance(struct Node *N) {
  if (N == NULL)
    return 0;
  return height(N->left) - height(N->right);
}

// Insert node
struct Node *insertNode(struct Node *node, int key) {
  // Find the correct position to insertNode the node and insertNode it
  if (node == NULL)
    return (newNode(key));

  if (key < node->key)
    node->left = insertNode(node->left, key);
  else if (key > node->key)
    node->right = insertNode(node->right, key);
  else
    return node;

  // Update the balance factor of each node and
  // Balance the tree
  node->height = 1 + max(height(node->left),
               height(node->right));

  int balance = getBalance(node);
  if (balance > 1 && key < node->left->key)
    return rightRotate(node);

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

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

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

  return node;
}

struct Node *minValueNode(struct Node *node) {
  struct Node *current = node;

  while (current->left != NULL)
    current = current->left;

  return current;
}

// Delete a nodes
struct Node *deleteNode(struct Node *root, int key) {
  // Find the node and delete it
  if (root == NULL)
    return root;

  if (key < root->key)
    root->left = deleteNode(root->left, key);

  else if (key > root->key)
    root->right = deleteNode(root->right, key);

  else {
    if ((root->left == NULL) || (root->right == NULL)) {
      struct Node *temp = root->left ? root->left : root->right;

      if (temp == NULL) {
        temp = root;
        root = NULL;
      } else
        *root = *temp;
      free(temp);
    } else {
      struct Node *temp = minValueNode(root->right);

      root->key = temp->key;

      root->right = deleteNode(root->right, temp->key);
    }
  }

  if (root == NULL)
    return root;

  // Update the balance factor of each node and
  // balance the tree
  root->height = 1 + max(height(root->left),
               height(root->right));

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

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

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

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

  return root;
}

// Print the tree
void printPreOrder(struct Node *root) {
  if (root != NULL) {
    printf("%d ", root->key);
    printPreOrder(root->left);
    printPreOrder(root->right);
  }
}

int main() {
  struct Node *root = NULL;

  root = insertNode(root, 2);
  root = insertNode(root, 1);
  root = insertNode(root, 7);
  root = insertNode(root, 4);
  root = insertNode(root, 5);
  root = insertNode(root, 3);
  root = insertNode(root, 8);

  printPreOrder(root);

  root = deleteNode(root, 3);

  printf("\nAfter deletion: ");
  printPreOrder(root);

  return 0;
}

C++

// AVL tree implementation in C++

#include <iostream>
using namespace std;

class Node {
   public:
  int key;
  Node *left;
  Node *right;
  int height;
};

int max(int a, int b);

// Calculate height
int height(Node *N) {
  if (N == NULL)
    return 0;
  return N->height;
}

int max(int a, int b) {
  return (a > b) ? a : b;
}

// New node creation
Node *newNode(int key) {
  Node *node = new Node();
  node->key = key;
  node->left = NULL;
  node->right = NULL;
  node->height = 1;
  return (node);
}

// Rotate right
Node *rightRotate(Node *y) {
  Node *x = y->left;
  Node *T2 = x->right;
  x->right = y;
  y->left = T2;
  y->height = max(height(y->left),
          height(y->right)) +
        1;
  x->height = max(height(x->left),
          height(x->right)) +
        1;
  return x;
}

// Rotate left
Node *leftRotate(Node *x) {
  Node *y = x->right;
  Node *T2 = y->left;
  y->left = x;
  x->right = T2;
  x->height = max(height(x->left),
          height(x->right)) +
        1;
  y->height = max(height(y->left),
          height(y->right)) +
        1;
  return y;
}

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

// Insert a node
Node *insertNode(Node *node, int key) {
  // Find the correct postion and insert the node
  if (node == NULL)
    return (newNode(key));
  if (key < node->key)
    node->left = insertNode(node->left, key);
  else if (key > node->key)
    node->right = insertNode(node->right, key);
  else
    return node;

  // Update the balance factor of each node and
  // balance the tree
  node->height = 1 + max(height(node->left),
               height(node->right));
  int balanceFactor = getBalanceFactor(node);
  if (balanceFactor > 1) {
    if (key < node->left->key) {
      return rightRotate(node);
    } else if (key > node->left->key) {
      node->left = leftRotate(node->left);
      return rightRotate(node);
    }
  }
  if (balanceFactor < -1) {
    if (key > node->right->key) {
      return leftRotate(node);
    } else if (key < node->right->key) {
      node->right = rightRotate(node->right);
      return leftRotate(node);
    }
  }
  return node;
}

// Node with minimum value
Node *nodeWithMimumValue(Node *node) {
  Node *current = node;
  while (current->left != NULL)
    current = current->left;
  return current;
}

// Delete a node
Node *deleteNode(Node *root, int key) {
  // Find the node and delete it
  if (root == NULL)
    return root;
  if (key < root->key)
    root->left = deleteNode(root->left, key);
  else if (key > root->key)
    root->right = deleteNode(root->right, key);
  else {
    if ((root->left == NULL) ||
      (root->right == NULL)) {
      Node *temp = root->left ? root->left : root->right;
      if (temp == NULL) {
        temp = root;
        root = NULL;
      } else
        *root = *temp;
      free(temp);
    } else {
      Node *temp = nodeWithMimumValue(root->right);
      root->key = temp->key;
      root->right = deleteNode(root->right,
                   temp->key);
    }
  }

  if (root == NULL)
    return root;

  // Update the balance factor of each node and
  // balance the tree
  root->height = 1 + max(height(root->left),
               height(root->right));
  int balanceFactor = getBalanceFactor(root);
  if (balanceFactor > 1) {
    if (getBalanceFactor(root->left) >= 0) {
      return rightRotate(root);
    } else {
      root->left = leftRotate(root->left);
      return rightRotate(root);
    }
  }
  if (balanceFactor < -1) {
    if (getBalanceFactor(root->right) <= 0) {
      return leftRotate(root);
    } else {
      root->right = rightRotate(root->right);
      return leftRotate(root);
    }
  }
  return root;
}

// Print the tree
void printTree(Node *root, string indent, bool last) {
  if (root != nullptr) {
    cout << indent;
    if (last) {
      cout << "R----";
      indent += "   ";
    } else {
      cout << "L----";
      indent += "|  ";
    }
    cout << root->key << endl;
    printTree(root->left, indent, false);
    printTree(root->right, indent, true);
  }
}

int main() {
  Node *root = NULL;
  root = insertNode(root, 33);
  root = insertNode(root, 13);
  root = insertNode(root, 53);
  root = insertNode(root, 9);
  root = insertNode(root, 21);
  root = insertNode(root, 61);
  root = insertNode(root, 8);
  root = insertNode(root, 11);
  printTree(root, "", true);
  root = deleteNode(root, 13);
  cout << "After deleting " << endl;
  printTree(root, "", true);
}

Complexities of Different Operations on an AVL Tree

nsertionDeletionSearch
O(log n)O(log n)O(log n)

AVL Tree Applications

For indexing large records in databases
For searching in large databases


Thanks for reading! We hope you found this tutorial helpful and we would love to hear your feedback in the Comments section below. And show us what you’ve learned by sharing your photos and creative projects with us.

salman khan

Written by worldofitech

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Binary Search Tree (BST)

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