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Parallel algorithms for constructing red–black trees from sorted lists of items can run in constant time or ( ) time, depending on the computer model, if the number of processors available is asymptotically proportional to the number of items where . Fast search, insertion, and deletion parallel algorithms are also known.
AA trees are named after their originator, Swedish computer scientist Arne Andersson. [1] AA trees are a variation of the red–black tree, a form of binary search tree which supports efficient addition and deletion of entries. Unlike red–black trees, red nodes on an AA tree can only be added as a right subchild.
All of the red-black tree algorithms that have been proposed are characterized by a worst-case search time bounded by a small constant multiple of log N in a tree of N keys, and the behavior observed in practice is typically that same multiple faster than the worst-case bound, close to the optimal log N nodes examined that would be observed in a perfectly balanced tree.
Fig. 1: A binary search tree of size 9 and depth 3, with 8 at the root. In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree data structure with the key of each internal node being greater than all the keys in the respective node's left subtree and less than the ones in its right subtree.
2–3–4 trees are B-trees of order 4; [1] like B-trees in general, they can search, insert and delete in O(log n) time.One property of a 2–3–4 tree is that all external nodes are at the same depth.
Both AVL trees and red–black (RB) trees are self-balancing binary search trees and they are related mathematically. Indeed, every AVL tree can be colored red–black, [14] but there are RB trees which are not AVL balanced. For maintaining the AVL (or RB) tree's invariants, rotations play an important role.
To turn a regular search tree into an order statistic tree, the nodes of the tree need to store one additional value, which is the size of the subtree rooted at that node (i.e., the number of nodes below it). All operations that modify the tree must adjust this information to preserve the invariant that size[x] = size[left[x]] + size[right[x]] + 1
A faster method (due to Floyd [8]) starts by arbitrarily putting the elements on a binary tree, respecting the shape property (the tree could be represented by an array, see below). Then starting from the lowest level and moving upwards, sift the root of each subtree downward as in the deletion algorithm until the heap property is restored.