Tree structure

A tree structure, tree diagram, or tree model is a way of representing the hierarchical nature of a structure in a graphical form. It is named a "tree structure" because the classic representation resembles a tree, although the chart is generally upside down compared to a biological tree, with the "stem" at the top and the "leaves" at the bottom.

A tree structure is conceptual, and appears in several forms. For a discussion of tree structures in specific fields, see Tree (data structure) for computer science; insofar as it relates to graph theory, see tree (graph theory) or tree (set theory). Other related articles are listed below.

Terminology and properties

The tree elements are called "nodes". The lines connecting elements are called "branches". Nodes without children are called leaf nodes, "end-nodes", or "leaves".

Every finite tree structure has a member that has no superior. This member is called the "root" or root node. The root is the starting node. But the converse is not true: infinite tree structures may or may not have a root node.

The names of relationships between nodes model the kinship terminology of family relations. The gender-neutral names "parent" and "child" have largely displaced the older "father" and "son" terminology. The term "uncle" is still widely used for other nodes at the same level as the parent, although it is sometimes replaced with gender-neutral terms like "ommer".^[1]

A node's "parent" is a node one step higher in the hierarchy (i.e. closer to the root node) and lying on the same branch.
"Sibling" ("brother" or "sister") nodes share the same parent node.
A node's "uncles" (sometimes "ommers") are siblings of that node's parent.
A node that is connected to all lower-level nodes is called an "ancestor". The connected lower-level nodes are "descendants" of the ancestor node.

In the example, "encyclopedia" is the parent of "science" and "culture", its children. "Art" and "craft" are siblings, and children of "culture", which is their parent and thus one of their ancestors. Also, "encyclopedia", as the root of the tree, is the ancestor of "science", "culture", "art" and "craft". Finally, "science", "art" and "craft", as leaves, are ancestors of no other node.

Tree structures can depict all kinds of taxonomic knowledge, such as family trees, the biological evolutionary tree, the evolutionary tree of a language family, the grammatical structure of a language (a key example being S → NP VP, meaning a sentence is a noun phrase and a verb phrase, with each in turn having other components which have other components), the way web pages are logically ordered in a web site, mathematical trees of integer sets, et cetera.

The Oxford English Dictionary records use of both the terms "tree structure" and "tree-diagram" from 1965 in Noam Chomsky's Aspects of the Theory of Syntax.^[2]

In a tree structure there is one and only one path from any point to any other point.

Computer science uses tree structures extensively (see Tree (data structure) and telecommunications.)

For a formal definition see set theory, and for a generalization in which children are not necessarily successors, see prefix order.

Examples of tree structures

A diagram in the shape of a tree illustrating the "evolution" of thermionic tubes (a type of vacuum tube) between 1883 and 1934

Internet:
- usenet hierarchy
- Document Object Model's logical structure,^[3] Yahoo! subject index, Curlie
Operating system: directory structure
Information management: Dewey Decimal System, PSH, this hierarchical bulleted list
Management: hierarchical organizational structures
Computer science:
Biology: evolutionary tree
Business: pyramid selling scheme
Project management: work breakdown structure
Linguistics:
- (Syntax) Phrase structure trees
- (Historical Linguistics) Tree model of language change
Sports: business chess, playoffs brackets
Mathematics: Von Neumann universe
Group theory: descendant trees

Representing trees

There are many ways of visually representing tree structures. Almost always, these boil down to variations, or combinations, of a few basic styles:

Classical node-link diagrams

Classical node-link diagrams, that connect nodes together with line segments:

encyclopedia
/ culture		\ science
/ art	\ craft

Nested sets

Nested sets that use enclosure or containment to show parenthood; examples include TreeMaps, fractal maps, and Euler diagrams:

encyclopedia

	culture
	culture
art craft

science

Layered "icicle" diagrams

Layered "icicle" diagrams that use alignment/adjacency.

encyclopedia
culture		science
art	craft

Outlines and tree views

Lists or diagrams that use indentation, sometimes called "outlines" or "tree views".

An outline:

encyclopedia

culture

art

craft

science

A tree view:

encyclopedia
- culture
  - art
  - craft
- science

Nested parentheses

A correspondence to nested parentheses was first noticed by Sir Arthur Cayley:

((art,craft)culture,science)encyclopedia
or
encyclopedia(culture(art,craft),science)

Radial trees

Trees can also be represented radially:

art \	craft /
culture \|
encyclopedia
\| science

References

^ "Ethereum Glossary". GitHub. Archived from the original on 25 April 2019. Retrieved 17 April 2019.
^ "tree". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
^ "What is the Document Object Model?". W3C Architecture domain. Archived from the original on 2012-02-12. Retrieved 2006-12-05.

External links

[1] "Ethereum Glossary". GitHub. Archived from the original on 25 April 2019. Retrieved 17 April 2019.

[2] "tree". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)

[3] "What is the Document Object Model?". W3C Architecture domain. Archived from the original on 2012-02-12. Retrieved 2006-12-05.

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