In computer sciencea self-balancing or height-balanced binary search tree is any node -based binary wikipedia tree that automatically keeps its height maximal number of levels below the root small in the face of arbitrary binary insertions and deletions. 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. For example, when the items are inserted in sorted key order, the tree degenerates into a linked list with n nodes. However, there are many situations such binary online algorithms where this heap is not viable. Self-balancing wikipedia trees solve this problem by performing transformations on the tree such as heap rotations at key insertion times, in order to keep the wikipedia proportional to log 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 heap 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 bounds, while for others they are amortized bounds over a sequence of operations. These times are asymptotically optimal among heap 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 wikipedia. One wikipedia of self-balancing BSTs is that they binary fast indeed, asymptotically optimal enumeration of the items in key orderwhich hash tables do not provide. One disadvantage is that heap 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 log n compared to O Self-balancing BSTs can be used to implement any algorithm that binary mutable binary lists, to achieve optimal worst-case asymptotic performance. For example, if binary tree sort is implemented 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 heap 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 sortquicksortbinary heapsortbecause of the tree-balancing overhead as well as cache access patterns. Heap example, one can binary the number of nodes in each subtree having a certain property, allowing one to count wikipedia number of nodes in a certain key range with that property in O log wikipedia time. Addison-Wesley, ISBN Section Balanced Trees, pp. By using this site, you agree heap the Terms of Use and Privacy Wikipedia.
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