In computer science, an AVL tree is a self-balancing binary search tree, and it was the first such data structure to be invented.

In other words AVL tree is a binary search tree where the height of the left subtree differs from the height of the right subtree by at most 1 level, if it exceeds 1 level then rebalancing occurs.

Basic operations of an AVL tree involve carrying out the same actions as would be carried out on an unbalanced binary search tree, but modifications are preceded or followed by one or more operations called tree rotations, which help to restore the height balance of the subtrees.

Lookup in an AVL tree is performed exactly like in any unbalanced binary search tree.

< center > Deleting a node with two children from a binary search tree </ center >

Both AVL trees and red-black trees are self-balancing binary search trees, so they are very similar mathematically.

For instance, binary search is said to run in a number of steps proportional to the logarithm of the length of the list being searched, or in O ( log ( n )), colloquially " in logarithmic time ".

For example, if the sorted list to which we apply binary search has n elements, and we can guarantee that each lookup of an element in the list can be done in unit time, then at most log < sub > 2 </ sub > n + 1 time units are needed to return an answer.

Take as an example a program that looks up a specific entry in a sorted list of size n. Suppose this program were implemented on Computer A, a state-of-the-art machine, using a linear search algorithm, and on Computer B, a much slower machine, using a binary search algorithm.

Elaborate schemes allowed users to download binary files, search gopherspace, and interact with distant programs, all using plain text e-mail.

In computer science, a binary search or half-interval search algorithm finds the position of a specified value ( the input " key ") within a sorted array.

A binary search halves the number of items to check with each iteration, so locating an item ( or determining its absence ) takes logarithmic time.

Binary trees are used to implement binary search trees and binary heaps.

Tree rotations are very common internal operations on Self-balancing binary search tree | self-balancing binary trees.

The B-tree is a generalization of a binary search tree in that a node can have more than two children.

Unlike self-balancing binary search trees, the B-tree is optimized for systems that read and write large blocks of data.

In automatic systems this can be done using a binary search algorithm or interpolation search ; manual searching may be performed using a roughly similar procedure, though this will often be done unconsciously.

The simplest form of index is a sorted list of values that can be searched using a binary search with an adjacent reference to the location of the entry, analogous to the index in the back of a book.

The binary abacus is used to explain how computers manipulate numbers.

** For every non-empty set S there is a binary operation defined on S that makes it a group.

( A cancellative binary operation is enough.

A simple form of AM, often used for digital communications, is on-off keying: a type of amplitude-shift keying in which binary data is represented by the presence or absence of a carrier.

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X

If there is a mixture of only two types of atoms, not counting impurities, such as a copper-nickel alloy, then it is called a binary alloy.

In this respect, all of the various forms of an alloy containing only two constituents, like iron and carbon, is called a binary system, while all of the alloy combinations possible with a ternary alloy, such as alloys of iron, carbon and chromium, is called a ternary system.

The observations show that the cloud is asymmetrical and matches the pattern of X-ray binaries ( binary star systems containing black holes or neutron stars ), mostly on one side of the galactic center.

In mathematics, the associative property is a property of some binary operations.

Formally, a binary operation on a set S is called associative if it satisfies the associative law:

* Absorbing element, in mathematics, an element that does not change when it is combined in a binary operation with some other element

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

As with all binary trees, a node's in-order successor is the left-most child of its right subtree, and a node's in-order predecessor is the right-most child of its left subtree.

The redesigned AIM for Mac is a full universal binary Cocoa API application that supports both Tiger and Leopard – Mac OS X 10. 4. 8 ( and above ) or Mac OS X 10. 5. 3 ( and above ).

However, Alpha Centauri is a bright binary star, whose unresolved components to the naked eye are both fainter than Arcturus.

Hipparcos also suggested that Arcturus is a binary star, with the companion about twenty times dimmer than the primary and orbiting close enough to be at the very limits of our current ability to make it out.

Prokaryotic cells, which lack a nucleus, divide by a process called binary fission.

This is because, for a perfect case, each intersection test would divide the possibilities by two, and we would have a binary tree type structure.

Bacteria divide asexually via binary fission ; viruses take control of host cells to produce more viruses ; Hydras ( invertebrates of the order Hydroidea ) and yeasts are able to reproduce by budding.

Proplastids and young chloroplasts commonly divide by binary fission, but more mature chloroplasts also have this capacity.

The name " divide and conquer " is sometimes applied also to algorithms that reduce each problem to only one subproblem, such as the binary search algorithm for finding a record in a sorted list ( or its analog in numerical computing, the bisection algorithm for root finding ).

However significant such changes may seem, it is arguable that the effective imposition of an average undergraduate fee of £ 7. 5 K pa for the three institutions ( and others, but not to Cardiff, Swansea, Bangor and Aberystwyth all of whom will charge £ 9 K pa ) will cause much more substantial long term damage to these universities and reinvent the ' binary divide ' between universities and the former polytechnics and HE institutes

An example of a " divide and conquer algorithm ", binary search involves each element being found by determining which half of the table a match may be found in and repeating until either success or failure.

Over millions of years the endosymbiotic cyanobacterium evolved structurally and functionally, retaining its own DNA and the ability to divide by binary fission ( not mitotically ) but giving up its autonomy by the transfer of some of its genes to the nuclear genome.

The cells divide by binary fission, rather than budding.

The microspheres were able to asexually divide via binary fission, could form junctions with other microspheres, and developed a double membrane corresponding to that of a cell.

* The Key To The Universe ( previously Numbers: The Key To The Universe, ISBN 0-439-98116-6 ( Fibonacci Series, Golden Ratio, properties of Square, Triangle, Cube, Centred Hexagon and Tetrahedral numbers, " difference of two squares ", number superstitions, prime numbers, Mersenne primes, tests to see if a number will divide by anything from 2-13 and 19, finger multiplication, binary and base 8, perfect numbers, irrational transcendental and imaginary numbers, infinity.