2-3 tree

# 2-3 tree

Discussion

Encyclopedia
In computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, a 2-3 tree is a type of data structure
Data structure
In computer science, a data structure is a particular way of storing and organizing data in a computer so that it can be used efficiently.Different kinds of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks...

, a tree
Tree (data structure)
In computer science, a tree is a widely-used data structure that emulates a hierarchical tree structure with a set of linked nodes.Mathematically, it is an ordered directed tree, more specifically an arborescence: an acyclic connected graph where each node has zero or more children nodes and at...

where every node
Node (computer science)
A node is a record consisting of one or more fields that are links to other nodes, and a data field. The link and data fields are often implemented by pointers or references although it is also quite common for the data to be embedded directly in the node. Nodes are used to build linked, often...

with children (internal node) has either two children (2-node) and one data element
Data element
In metadata, the term data element is an atomic unit of data that has precise meaning or precise semantics. A data element has:# An identification such as a data element name# A clear data element definition# One or more representation terms...

or three children (3-nodes) and two data elements. Nodes on the outside of the tree (leaf nodes) have no children and one or two data elements.

2-3 trees are an isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

of AA tree
AA tree
An AA tree in computer science is a form of balanced tree used for storing and retrieving ordered data efficiently. AA trees are named for Arne Andersson, their inventor....

s, meaning that they are equivalent data structures. In other words, for every 2-3 tree, there exists at least one AA tree with data elements in the same order. 2-3 trees are balanced, meaning that each right, center, and left subtree contains the same or close to the same amount of data.

## Properties

• Every non-leaf is a 2-node or a 3-node. A 2-node contains one data item and has two children. A 3-node contains two data items and has 3 children.
• All leaves are at the same level (the bottom level)
• All data are kept in sorted order
• Every leaf node will contain 1 or 2 fields.

## Non-leaf nodes

These contain one or two fields which indicate the range of values in its subtrees. If a node has two children, it will have one field; if the node has three children, it will have two fields. Each non-leaf node will contain a value in field 1 which is greater than the largest item in its left sub-tree, but less than or equal to the smallest item in its right sub-tree (or center sub-tree, if it has three children). If that node has three children, field 2 contains a value which is greater than the largest value in the center sub-tree, but less than or equal to the smallest item in its right sub-tree. The purpose of these values is to direct a search function to the correct sub-tree and eventually to the correct data node.