In computer science, a self-balancing binary search tree (BST) 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.[1] These operations when designed for a self-balancing binary search tree, contain precautionary measures against boundlessly increasing tree height, so that these abstract data structures receive the attribute "self-balancing".
For height-balanced binary trees, the height is defined to be logarithmic
O(logn)
n
Self-balancing binary search trees provide efficient implementations for mutable ordered lists, and can be used for other abstract data structures such as associative arrays, priority queues and sets.
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 20+21+···+2h = 2h+1−1 nodes. It follows that for any tree with n nodes and height h:
n\le2h+1-1
And that implies:
h\ge\lceillog2(n+1)-1\rceil\ge\lfloorlog2n\rfloor
In other words, the minimum height of a binary tree with n nodes is rounded down; that is,
\lfloorlog2n\rfloor
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: for example, when n = 1,000,000, the minimum height is
\lfloorlog2(1,000,000)\rfloor=19
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 Although a certain overhead is involved, it is not bigger than the always necessary lookup cost and may be justified by ensuring fast execution of all operations.
While it is possible to maintain a BST with minimum height with expected
O(logn)
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(logn)
O(n)
Data structures implementing this type of tree include:
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 (indeed, asymptotically optimal) enumeration of the items in key order, which hash tables do not provide. One disadvantage is that their 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 most[2] hash tables (
O(logn)
O(n)
O(logn)
O(1)
O(nlogn)
Self-balancing BSTs are flexible data structures, in that it's easy to extend them to efficiently record 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 number of nodes in a certain key range with that property in
O(logn)