## Self-balancing binary search tree

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In computer sciencea self-balancing or height-balanced binary search tree is any node -based binary search tree that automatically keeps its height maximal number of levels below the root small in the face of arbitrary item insertions and deletions. These structures provide efficient implementations for mutable ordered listsand can be used for other abstract data structures such as associative arrayspriority queues and sets. The red—black treewhich is a type of self-balancing binary search tree, was called symmetric binary B-tree [2] and was renamed but can still be confused with perfect binary search tree generic concept of self-balancing binary search tree because of the initials.

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. It follows that for a tree with n nodes and height h:.

However, the simplest algorithms for BST item insertion may yield a tree with height n in rather common situations. For example, when the items are inserted in sorted key order, the tree degenerates into a linked list with n nodes.

The difference in performance between the two situations may be enormous: If the data items are known ahead of time, the height can be kept small, in the average sense, by adding values in a random order, resulting in a random binary search tree. However, there are many situations such as online algorithms where this randomization is not viable. Self-balancing binary trees solve this problem by performing transformations on the tree such as tree rotations at key insertion times, in order to keep the height proportional to log 2 n.

Although a certain overhead is involved, it may be justified in the long run by ensuring fast execution of later operations. In the asymptotic " Big-O " sense, a self-balancing BST structure containing n items allows the lookup, insertion, and removal of an item in O log n worst-case time, and ordered enumeration of all items in O n time.

For some implementations these are per-operation time perfect binary search tree, while for others they are amortized bounds over a sequence of operations. These times are asymptotically optimal among all data structures that manipulate the key only through comparisons.

Self-balancing binary search trees can be used in a natural way to construct and maintain ordered lists, such as priority queues. They can also be used for associative arrays ; key-value pairs are simply inserted with an ordering based on the key alone. In this capacity, self-balancing BSTs have a number of advantages and disadvantages over their main competitor, hash tables.

One advantage of self-balancing BSTs is that they allow fast perfect binary search tree, asymptotically optimal enumeration of the items in key orderwhich hash tables do not provide. One disadvantage is that perfect binary search tree lookup algorithms get more complicated when there may be multiple items with the same key. Self-balancing BSTs have better worst-case lookup performance than hash tables O log n compared to O nbut have worse average-case performance O **perfect binary search tree** n compared to O 1.

Self-balancing BSTs can be used to implement any algorithm that requires mutable ordered lists, to achieve optimal worst-case asymptotic performance. For example, if binary tree sort is perfect binary search tree with a self-balanced BST, we have a very simple-to-describe yet asymptotically optimal O n log n sorting algorithm.

Similarly, many algorithms in computational geometry exploit variations on self-balancing BSTs to solve problems such as the line segment intersection problem and the point location problem efficiently. For average-case performance, however, self-balanced BSTs may be less efficient than other solutions. Binary tree sort, in particular, is likely to be slower than merge sortquicksortor heapsortbecause of the tree-balancing overhead as well as cache access patterns.

Self-balancing BSTs are flexible data structures, in that it's easy to extend them to efficiently perfect binary search tree additional information or perform new operations. For example, one can record the number of nodes in each subtree having a certain property, allowing one to count the perfect binary search tree of nodes in a certain key range with that property in O log n time. These extensions can be used, for example, to optimize database queries or other list-processing algorithms.

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November Learn how and when to remove this template message. The Art of Computer ProgrammingVolume 3: Sorting and SearchingSecond Edition. Binary decision diagram Directed acyclic graph Directed acyclic word graph. List of data structures.

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