# Arborescence (graph theory)

In graph theory, an **arborescence** is a directed graph in which, for a vertex *u* called the root and any other vertex *v*, there is exactly one directed path from *u* to *v*. An arborescence is thus the directed-graph form of a rooted tree, understood here as an undirected graph.[1][2]

Equivalently, an arborescence is a directed, rooted tree in which all edges point away from the root; a number of other equivalent characterizations exist.[3][4] Every arborescence is a directed acyclic graph (DAG), but not every DAG is an arborescence.

An arborescence can equivalently be defined as a rooted digraph in which the path from the root to any other vertex is unique.[1]

The term *arborescence* comes from French.[5] Some authors object to it on grounds that it is cumbersome to spell.[6] There is a large number of synonyms for arborescence in graph theory, including **directed rooted tree**[2][6] **out-arborescence**,[7] **out-tree**,[8] and even **branching** being used to denote the same concept.[8] *Rooted tree* itself has been defined by some authors as a directed graph.[9][10][11]

Furthermore, some authors define an arborescence to be a spanning directed tree of a given digraph.[11][12] The same can be said about some its synonyms, especially *branching*.[12] Other authors use *branching* to denote a forest of arborescences, with the latter notion defined in broader sense given at beginning of this article,[13][14] but a variation with both notions of the spanning flavor is also encountered.[11]

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**[15] or **anti-arborescence**[16] etc. W. T. Tutte distinguishes between the two cases by using the phrases *arborescence diverging from* [some root] and *arborescence converging to* [some root].[17]

The number of rooted trees (or arborescences) with *n* nodes is given by the sequence:

## See also

## References

- Gordon, Gary (1989). "A greedoid polynomial which distinguishes rooted arborescences".
*Proceedings of the American Mathematical Society*.**107**(2): 287. doi:10.1090/S0002-9939-1989-0967486-0. - Stanley Gill Williamson (1985).
*Combinatorics for Computer Science*. Courier Dover Publications. p. 288. ISBN 978-0-486-42076-9. - Jean-Claude Fournier (2013).
*Graphs Theory and Applications: With Exercises and Problems*. John Wiley & Sons. pp. 94–95. ISBN 978-1-84821-070-7. - Jean Gallier (2011).
*Discrete Mathematics*. Springer Science & Business Media. pp. 193–194. ISBN 978-1-4419-8046-5. - Per Hage and Frank Harary (1996).
*Island Networks: Communication, Kinship, and Classification Structures in Oceania*. Cambridge University Press. p. 43. ISBN 978-0-521-55232-5. - Mehran Mesbahi; Magnus Egerstedt (2010).
*Graph Theoretic Methods in Multiagent Networks*. Princeton University Press. p. 38. ISBN 1-4008-3535-6. - Ding-Zhu Du; Ker-I Ko; Xiaodong Hu (2011).
*Design and Analysis of Approximation Algorithms*. Springer Science & Business Media. p. 108. ISBN 978-1-4614-1701-9. - Jonathan L. Gross; Jay Yellen; Ping Zhang (2013).
*Handbook of Graph Theory, Second Edition*. CRC Press. p. 116. ISBN 978-1-4398-8018-0. - David Makinson (2012).
*Sets, Logic and Maths for Computing*. Springer Science & Business Media. pp. 167–168. ISBN 978-1-4471-2499-3. - Kenneth Rosen (2011).
*Discrete Mathematics and Its Applications, 7th edition*. McGraw-Hill Science. p. 747. ISBN 978-0-07-338309-5. - Alexander Schrijver (2003).
*Combinatorial Optimization: Polyhedra and Efficiency*. Springer. p. 34. ISBN 3-540-44389-4. - Jørgen Bang-Jensen; Gregory Z. Gutin (2008).
*Digraphs: Theory, Algorithms and Applications*. Springer. p. 339. ISBN 978-1-84800-998-1. - James Evans (1992).
*Optimization Algorithms for Networks and Graphs, Second Edition*. CRC Press. p. 59. ISBN 978-0-8247-8602-1. - Bernhard Korte; Jens Vygen (2012).
*Combinatorial Optimization: Theory and Algorithms*(5th ed.). Springer Science & Business Media. p. 18. ISBN 978-3-642-24488-9. - Kurt Mehlhorn; Peter Sanders (2008).
*Algorithms and Data Structures: The Basic Toolbox*(PDF). Springer Science & Business Media. p. 52. ISBN 978-3-540-77978-0. - Bernhard Korte; Jens Vygen (2012).
*Combinatorial Optimization: Theory and Algorithms*(5th ed.). Springer Science & Business Media. p. 28. ISBN 978-3-642-24488-9. - Tutte, W.T. (2001),
*Graph Theory*, Cambridge University Press, pp. 126–127, ISBN 978-0-521-79489-3

## External links