Avl Trees

Introduction

AVL Trees, named after inventors Adelson-Velsky and Landis, are self-balancing binary search trees. In an AVL tree, the heights of the two child subtrees of any node differ by no more than one. If at any time they differ by more than one, rebalancing is done to restore this property.

Explanation

Rebalancing is done through rotations - single or double. The four possible rotations are left rotation, right rotation, left-right rotation, and right-left rotation. This ensures that the tree remains balanced, thereby guaranteeing (O(\log n)) time complexity for all basic dynamic set operations.

Basic Operations

1. Insertion:

   • Similar to a regular BST insertion, but followed by checking and adjusting the balance factor (the difference in height between the left and right subtree) of each node, starting from the newly inserted node up to the root.

2. Deletion:

   • After deletion, similar to insertion, the tree is traversed upwards from the deleted node, and rotations are performed where necessary to maintain the tree’s balance.

3. Searching:

   • Searching in an AVL tree is identical to searching in a BST, as the AVL property doesn’t affect the order of the elements.

JavaScript Implementation

Below is a simplified example showing the structure of an AVL Tree class. The actual implementation of rotations and balancing is complex and involves multiple cases:

	class AVLNode extends Node {
	  constructor(data) {
	    super(data);
	    this.height = 1; // New property for balancing
	  }
	}
	
	class AVLTree extends BinarySearchTree {
	  constructor() {
	    super();
	  }
	
	  // Insertion with balancing
	  insert(data) {
	    this.root = this.insertNode(this.root, data);
	    // Balancing logic to be applied after insertion
	  }
	
	  // Deletion with balancing
	  delete(data) {
	    this.root = this.deleteNode(this.root, data);
	    // Balancing logic to be applied after deletion
	  }
	
	  // Method to update the height of a node
	  updateHeight(node) {
	    // Implementation to update the height
	  }
	
	  // Method to get the balance factor of a node
	  getBalanceFactor(node) {
	    // Implementation to get the balance factor
	  }
	
	  // Rotations (left, right, left-right, right-left)
	  // Implementations for rotations to maintain balance
	}
	
	// Example Usage
	const avlTree = new AVLTree();
	avlTree.insert(10);
	avlTree.insert(20);
	avlTree.insert(30);
	// Additional insertions and deletions

Pros and Cons

• Pros:

  • Maintains (O(\log n)) complexity for insertions, deletions, and searches by self-balancing.
  • More consistent performance than a regular BST, especially for datasets that undergo frequent insertions and deletions.

• Cons:

  • More complex to implement than a regular BST.
  • Slightly slower operations than a regular BST due to the overhead of balancing.

Practical Use Cases

1. Database Systems:

   • AVL Trees are used in database systems where quick retrieval, insertion, and deletion are critical, and data is frequently updated.

2. Memory Management:

   • Used in memory management subsystems of operating systems for efficient allocation and deallocation of memory.

Conclusion

AVL Trees are an advanced type of BST that ensure self-balancing to maintain optimal operation times. They are particularly useful in applications where the dataset is dynamic and requires frequent and efficient updates.