# Tree structure

A tree structure showing the possible hierarchical organization of an encyclopedia.
The original Encyclopédie used a tree diagram to show the way in which its subjects were ordered.

A tree structure or tree diagram 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, even though the chart is generally upside down compared to an actual tree, with the "root" 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 also tree (set theory). Other related pages are listed.

## 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, although the term "uncle" is still used for other nodes at the same level as the parent.

• 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" 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.[1]

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 tree map used to represent a directory structure as a nested set.

## 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, that connect nodes together with line segments:

encyclopedia
/
culture
\
science
/
art
\
craft

### Nested sets

Nested sets that use enclosure/containment to show parenthood, examples include TreeMaps and fractal maps:

encyclopedia
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)

Trees can also be represented radially:

art
\
craft
/
culture
|
encyclopedia
|
science