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With the new operations, the implementation of AVL trees can be more efficient and highly-parallelizable. [13] The function Join on two AVL trees t 1 and t 2 and a key k will return a tree containing all elements in t 1, t 2 as well as k. It requires k to be greater than all keys in t 1 and smaller than all keys in t 2.
Various height-balanced binary search trees were introduced to confine the tree height, such as AVL trees, Treaps, and red–black trees. [5] The AVL tree was invented by Georgy Adelson-Velsky and Evgenii Landis in 1962 for the efficient organization of information. [6] [7] It was the first self-balancing binary search tree to be invented. [8]
Most operations on a binary search tree (BST) take time directly proportional to the height of the tree, so it is desirable to keep the height small. A binary tree with height h can contain at most 2 0 +2 1 +···+2 h = 2 h+1 −1 nodes. It follows that for any tree with n nodes and height h: + And that implies:
In depth-first search (DFS), the search tree is deepened as much as possible before going to the next sibling. To traverse binary trees with depth-first search, perform the following operations at each node: [3] [4] If the current node is empty then return. Execute the following three operations in a certain order: [5] N: Visit the current node.
WAVL trees are named after AVL trees, another type of balanced search tree, and are closely related both to AVL trees and red–black trees, which all fall into a common framework of rank balanced trees. Like other balanced binary search trees, WAVL trees can handle insertion, deletion, and search operations in time O(log n) per operation. [1] [2]
In 2016, Blelloch et al. formally proposed the join-based algorithms, and formalized the join algorithm for four different balancing schemes: AVL trees, red–black trees, weight-balanced trees and treaps. In the same work they proved that Adams' algorithms on union, intersection and difference are work-optimal on all the four balancing schemes.
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Various search-tree data structures exist, several of which also allow efficient insertion and deletion of elements, which operations then have to maintain tree balance. Search trees are often used to implement an associative array. The search tree algorithm uses the key from the key–value pair to find a location, and then the application ...