AVL Tree
Introduction
An AVL tree is a type of binary search tree that was invented by two Soviet mathematicians, G.M. Adelson-Velsky and E.M. Landis. Named after its inventors, AVL trees are widely recognized in the field of computer science for their ability to self-balance, which ensures optimal efficiency in performing various operations such as searching, insertion, and deletion.
History
The AVL tree was first introduced in 1962 by Georgy Maximovich Adelson-Velsky and Evgenii Mikhailovich Landis. Their work marked a significant contribution to the field of computer science, particularly in the area of data structure. The AVL tree was the first data structure of its kind to be invented, paving the way for the development of other self-balancing binary search trees.
Characteristics
The primary characteristic of an AVL tree is its self-balancing property. This means that for every node in the tree, the height of the left and right subtrees differ by at most one. This characteristic is crucial as it ensures that the tree remains approximately balanced at all times, preventing the formation of a degenerate or pathological tree, which would resemble a linked list.
Operations
AVL trees support several operations, including searching, insertion, and deletion. These operations are carried out efficiently due to the self-balancing nature of the tree.
Searching
Searching in an AVL tree is similar to searching in a binary search tree. The search begins at the root node and traverses down the tree, comparing the search key with the keys of the nodes. The time complexity of this operation is O(log n), where n is the number of nodes in the tree.
Insertion
Insertion in an AVL tree involves two steps: the standard binary search tree insertion, and a balancing step. After inserting a node, the tree may become unbalanced. To restore the balance, a rotation operation is performed. There are four types of rotations: left-left, right-right, left-right, and right-left.
Deletion
Deletion in an AVL tree also involves the binary search tree deletion and a balancing step. After deleting a node, the tree may become unbalanced, and a rotation operation is required to restore the balance.
Rotations
Rotations are fundamental operations in AVL trees that are used to maintain the tree's balance. There are four types of rotations: left-left, right-right, left-right, and right-left. The names of the rotations indicate the unbalanced state of the tree.
Left-Left Rotation
A left-left rotation is performed when the left subtree of the left child of a node is longer than its right subtree. The rotation involves moving the parent node down and to the right, and moving the left child up and to the right.
Right-Right Rotation
A right-right rotation is performed when the right subtree of the right child of a node is longer than its left subtree. The rotation involves moving the parent node down and to the left, and moving the right child up and to the left.
Left-Right Rotation
A left-right rotation is a combination of a left rotation followed by a right rotation. It is performed when the right subtree of the left child of a node is longer than its left subtree.
Right-Left Rotation
A right-left rotation is a combination of a right rotation followed by a left rotation. It is performed when the left subtree of the right child of a node is longer than its right subtree.
Applications
AVL trees are used in various applications where balanced binary search trees are required. They are used in databases to maintain indexes, in file systems, and in memory allocation systems. AVL trees are also used in the implementation of certain algorithms in the field of network routing and data compression.