In
graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, an arborescence is a
directed graph in which, for a
vertex (called the ''root'') and any other vertex , there is exactly one
directed path from to .
An arborescence is thus the directed-graph form of a
rooted tree, understood here as an
undirected graph.
Equivalently, an arborescence is a directed, rooted
tree
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
in which all edges point away from the root; a number of other equivalent characterizations exist.
Every arborescence is a
directed acyclic graph
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one v ...
(DAG), but not every DAG is an arborescence.
An arborescence can equivalently be defined as a
rooted digraph In mathematics, and, in particular, in graph theory, a rooted graph is a graph in which one vertex has been distinguished as the root. Both directed and undirected versions of rooted graphs have been studied, and there are also variant definitions ...
in which the path from the root to any other vertex is unique.
Definition
The term ''arborescence'' comes from French.
Some authors object to it on grounds that it is cumbersome to spell.
There is a large number of synonyms for arborescence in graph theory, including directed rooted tree
out-arborescence,
out-tree,
and even branching being used to denote the same concept.
''Rooted tree'' itself has been defined by some authors as a directed graph.
Further Definitions
Furthermore, some authors define an arborescence to be a spanning directed tree of a given digraph.
The same can be said about some of its synonyms, especially ''branching''.
Other authors use ''branching'' to denote a forest of arborescences, with the latter notion defined in broader sense given at beginning of this article,
but a variation with both notions of the spanning flavor is also encountered.
It's also possible to define a useful notion by reversing all the arcs of an arborescence, i.e. making them all point to the root rather than away from it. Such digraphs are also designated by a variety of terms such as in-tree
or anti-arborescence
etc.
W. T. Tutte distinguishes between the two cases by using the phrases ''arborescence diverging from''
ome root
Ome may refer to:
Places
* Ome (Bora Bora), a public island in the lagoon of Bora Bora
* Ome, Lombardy, Italy, a town and ''comune'' in the Province of Brescia
* Ōme, Tokyo, a city in the Prefecture of Tokyo
* Ome (crater), a crater on Mars
Tran ...
and ''arborescence converging to''
ome root
Ome may refer to:
Places
* Ome (Bora Bora), a public island in the lagoon of Bora Bora
* Ome, Lombardy, Italy, a town and ''comune'' in the Province of Brescia
* Ōme, Tokyo, a city in the Prefecture of Tokyo
* Ome (crater), a crater on Mars
Tran ...
The number of rooted trees (or arborescences) with ''n'' nodes is given by the sequence:
: 0, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, ... .
See also
*
Edmonds' algorithm
*
Multitree
References
External links
*
*
Trees (graph theory)
Directed acyclic graphs
{{combin-stub