Editing Associative array (section)

=== Tree implementations ===
{{main | Search tree }}

==== Self-balancing binary search trees ====
Another common approach is to implement an associative array with a [[self-balancing binary search tree]], such as an [[AVL tree]] or a [[red–black tree]].<ref>
Joel Adams and Larry Nyhoff.
[http://cs.calvin.edu/books/c++/intro/3e/WebItems/Ch15-Web/STL-Trees.pdf "Trees in STL"].
Quote:

"The Standard Template library ... some of its containers -- the set<T>, map<T1, T2>, multiset<T>, and multimap<T1, T2> templates -- are generally built using a special kind of ''self-balancing binary search tree'' called a ''red–black tree''."
</ref>

Compared to hash tables, these structures have both strengths and weaknesses. The worst-case performance of self-balancing binary search trees is significantly better than that of a hash table, with a time complexity in [[big O notation]] of O(log ''n''). This is in contrast to hash tables, whose worst-case performance involves all elements sharing a single bucket, resulting in O(''n'') time complexity. In addition, and like all binary search trees, self-balancing binary search trees keep their elements in order. Thus, traversing its elements follows a least-to-greatest pattern, whereas traversing a hash table can result in elements being in seemingly random order. Because they are in order, tree-based maps can also satisfy range queries (find all values between two bounds) whereas a hashmap can only find exact values. However, hash tables have a much better average-case time complexity than self-balancing binary search trees of O(1), and their worst-case performance is highly unlikely when a good [[hash function]] is used.

A self-balancing binary search tree can be used to implement the buckets for a hash table that uses separate chaining. This allows for average-case constant lookup, but assures a worst-case performance of O(log ''n''). However, this introduces extra complexity into the implementation and may cause even worse performance for smaller hash tables, where the time spent inserting into and balancing the tree is greater than the time needed to perform a [[linear search]] on all elements of a linked list or similar data structure.<ref name="knuth">{{cite book| first=Donald |last=Knuth |author1-link=Donald Knuth| title = The Art of Computer Programming| volume = 3: ''Sorting and Searching''| edition = 2nd| publisher = Addison-Wesley| year = 1998| isbn = 0-201-89685-0| pages = 513–558}}</ref><ref>{{cite web |url=https://schani.wordpress.com/2010/04/30/linear-vs-binary-search/ |title=Linear vs Binary Search |last=Probst |first=Mark |date=2010-04-30 |access-date=2016-11-20 }}</ref>

==== Other trees ====
Associative arrays may also be stored in unbalanced [[binary search tree]]s or in data structures specialized to a particular type of keys such as [[radix tree]]s, [[trie]]s, [[Judy array]]s, or [[van Emde Boas tree]]s, though the relative performance of these implementations varies. For instance, Judy trees have been found to perform less efficiently than hash tables, while carefully selected hash tables generally perform more efficiently than adaptive radix trees, with potentially greater restrictions on the data types they can handle.<ref>{{Cite book|last1=Alvarez|first1=Victor|last2=Richter|first2=Stefan|last3=Chen|first3=Xiao|last4=Dittrich|first4=Jens|title=2015 IEEE 31st International Conference on Data Engineering |chapter=A comparison of adaptive radix trees and hash tables |date=April 2015|location=Seoul, South Korea|publisher=IEEE|pages=1227–1238|doi=10.1109/ICDE.2015.7113370|isbn=978-1-4799-7964-6|s2cid=17170456}}</ref> The advantages of these alternative structures come from their ability to handle additional associative array operations, such as finding the mapping whose key is the closest to a queried key when the query is absent in the set of mappings.