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In computer science, an AVL tree is a self-balancing binary search tree. It was the first such data structure to be invented. In an AVL tree, the heights of the two child subtrees of any node differ by at most one; if at any time they differ by more than one, rebalancing is done to restore this property. Lookup, insertion, and deletion all take O(log n) time in both the average and worst cases, where is the number of nodes in the tree prior to the operation. Insertions and deletions may require the tree to be rebalanced by one or more tree rotations.
In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree whose internal nodes each store a key greater than all the keys in the node's left subtree and less than those in its right subtree. A binary tree is a type of data structure for storing data such as numbers in an organized way. Binary search trees allow binary search for fast lookup, addition and removal of data items, and can be used to implement dynamic sets and lookup tables. The order of nodes in a BST means that each comparison skips about half of the remaining tree, so the whole lookup takes time proportional to the binary logarithm of the number of items stored in the tree. This is much better than the linear time required to find items by key in an (unsorted) array, but slower than the corresponding operations on hash tables. Several variants of the binary search tree have been studied.
In computer science, a binary tree is a tree data structure in which each node has at most two children, which are referred to as the left child and the right child. A recursive definition using just set theory notions is that a (non-empty) binary tree is a tuple, where L and R are binary trees or the empty set and S is a singleton set containing the root. Some authors allow the binary tree to be the empty set as well.
In computer science, a B-tree is a self-balancing tree data structure that maintains sorted data and allows searches, sequential access, insertions, and deletions in logarithmic time. The B-tree generalizes the binary search tree, allowing for nodes with more than two children. Unlike other self-balancing binary search trees, the B-tree is well suited for storage systems that read and write relatively large blocks of data, such as disks. It is commonly used in databases and file systems.
In computer science, a red–black tree is a kind of self-balancing binary search tree. Each node stores an extra bit representing "color", used to ensure that the tree remains balanced during insertions and deletions.
A splay tree is a binary search tree with the additional property that recently accessed elements are quick to access again. Like self-balancing binary search trees, a splay tree performs basic operations such as insertion, look-up and removal in O(log n) amortized time. For many sequences of non-random operations, splay trees perform better than other search trees, even performing better than O(log n) for sufficiently non-random patterns, all without requiring advance knowledge of the pattern. The splay tree was invented by Daniel Sleator and Robert Tarjan in 1985.
In discrete mathematics, tree rotation is an operation on a binary tree that changes the structure without interfering with the order of the elements. A tree rotation moves one node up in the tree and one node down. It is used to change the shape of the tree, and in particular to decrease its height by moving smaller subtrees down and larger subtrees up, resulting in improved performance of many tree operations.
In computer science, the treap and the randomized binary search tree are two closely related forms of binary search tree data structures that maintain a dynamic set of ordered keys and allow binary searches among the keys. After any sequence of insertions and deletions of keys, the shape of the tree is a random variable with the same probability distribution as a random binary tree; in particular, with high probability its height is proportional to the logarithm of the number of keys, so that each search, insertion, or deletion operation takes logarithmic time to perform.
In computer science, tree traversal is a form of graph traversal and refers to the process of visiting each node in a tree data structure, exactly once. Such traversals are classified by the order in which the nodes are visited. The following algorithms are described for a binary tree, but they may be generalized to other trees as well.
In computer science, a 2–3 tree is a tree data structure, where every node with children has either two children (2-node) and one data element or three children (3-nodes) and two data elements. A 2–3 tree is a B-tree of order 3. Nodes on the outside of the tree have no children and one or two data elements. 2–3 trees were invented by John Hopcroft in 1970.
In computer science, a scapegoat tree is a self-balancing binary search tree, invented by Arne Andersson and again by Igal Galperin and Ronald L. Rivest. It provides worst-case O(log n) lookup time, and O(log n) amortized insertion and deletion time.
An AA tree in computer science is a form of balanced tree used for storing and retrieving ordered data efficiently. AA trees are named for Arne Andersson, their inventor.
In computer science, a k-d tree is a space-partitioning data structure for organizing points in a k-dimensional space. k-d trees are a useful data structure for several applications, such as searches involving a multidimensional search key and creating point clouds. k-d trees are a special case of binary space partitioning trees.
In graph theory, an m-ary tree is a rooted tree in which each node has no more than m children. A binary tree is the special case where m = 2, and a ternary tree is another case with m = 3 that limits its children to three.
In computer science, a leftist tree or leftist heap is a priority queue implemented with a variant of a binary heap. Every node x has an s-value which is the distance to the nearest leaf in subtree rooted at x. In contrast to a binary heap, a leftist tree attempts to be very unbalanced. In addition to the heap property, leftist trees are maintained so the right descendant of each node has the lower s-value.
In computer science, weight-balanced binary trees (WBTs) are a type of self-balancing binary search trees that can be used to implement dynamic sets, dictionaries (maps) and sequences. These trees were introduced by Nievergelt and Reingold in the 1970s as trees of bounded balance, or BB[α] trees. Their more common name is due to Knuth.
A tango tree is a type of binary search tree proposed by Erik D. Demaine, Dion Harmon, John Iacono, and Mihai Pătrașcu in 2004. It is named after Buenos Aires, of which the tango is emblematic.
Right rotation refers to the following
In computer science, a WAVL tree or weak AVL tree is a self-balancing binary search tree. 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.
In computer science, a weak heap is a data structure for priority queues, combining features of the binary heap and binomial heap. It can be stored in an array as an implicit binary tree like a binary heap, and has the efficiency guarantees of binomial heaps.
References
- Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein, 16 July 2001, Introduction to Algorithms, second edition. McGraw-Hill, ISBN 0-07-013151-1. Chapter 13.