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Tree (graph theory)

Trees
A labeled tree with 6 vertices and 5 edges.
Verticesv
Edgesv − 1
Chromatic number2 if v > 1
Table of graphs and parameters

In graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph.[1] A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees.[2]

A directed tree,[3] oriented tree,[4][5] polytree,[6] or singly connected network[7] is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest.

The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree,[8][9] either making all its edges point away from the root—in which case it is called an arborescence[3][10] or out-tree[11][12]—or making all its edges point towards the root—in which case it is called an anti-arborescence[13] or in-tree.[11][14] A rooted tree itself has been defined by some authors as a directed graph.[15][16][17] A rooted forest is a disjoint union of rooted trees. A rooted forest may be directed, called a directed rooted forest, either making all its edges point away from the root in each rooted tree—in which case it is called a branching or out-forest—or making all its edges point towards the root in each rooted tree—in which case it is called an anti-branching or in-forest.

The term tree was coined in 1857 by the British mathematician Arthur Cayley.[18]

  1. ^ Bender & Williamson 2010, p. 171.
  2. ^ Bender & Williamson 2010, p. 172.
  3. ^ a b Deo 1974, p. 206.
  4. ^ See Harary & Sumner (1980).
  5. ^ See Simion (1991).
  6. ^ See Dasgupta (1999).
  7. ^ See Kim & Pearl (1983).
  8. ^ Stanley Gill Williamson (1985). Combinatorics for Computer Science. Courier Dover Publications. p. 288. ISBN 978-0-486-42076-9.
  9. ^ Mehran Mesbahi; Magnus Egerstedt (2010). Graph Theoretic Methods in Multiagent Networks. Princeton University Press. p. 38. ISBN 978-1-4008-3535-5.
  10. ^ 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.
  11. ^ a b Deo 1974, p. 207.
  12. ^ Jonathan L. Gross; Jay Yellen; Ping Zhang (2013). Handbook of Graph Theory, Second Edition. CRC Press. p. 116. ISBN 978-1-4398-8018-0.
  13. ^ Bernhard Korte; Jens Vygen (2012). Combinatorial Optimization: Theory and Algorithms (5th ed.). Springer Science & Business Media. p. 28. ISBN 978-3-642-24488-9.
  14. ^ 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. Archived (PDF) from the original on 2015-09-08.
  15. ^ David Makinson (2012). Sets, Logic and Maths for Computing. Springer Science & Business Media. pp. 167–168. ISBN 978-1-4471-2499-3.
  16. ^ Kenneth Rosen (2011). Discrete Mathematics and Its Applications, 7th edition. McGraw-Hill Science. p. 747. ISBN 978-0-07-338309-5.
  17. ^ Alexander Schrijver (2003). Combinatorial Optimization: Polyhedra and Efficiency. Springer. p. 34. ISBN 3-540-44389-4.
  18. ^ Cayley (1857) "On the theory of the analytical forms called trees," Philosophical Magazine, 4th series, 13 : 172–176.
    However it should be mentioned that in 1847, K.G.C. von Staudt, in his book Geometrie der Lage (Nürnberg, (Germany): Bauer und Raspe, 1847), presented a proof of Euler's polyhedron theorem which relies on trees on pages 20–21. Also in 1847, the German physicist Gustav Kirchhoff investigated electrical circuits and found a relation between the number (n) of wires/resistors (branches), the number (m) of junctions (vertices), and the number (μ) of loops (faces) in the circuit. He proved the relation via an argument relying on trees. See: Kirchhoff, G. R. (1847) "Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird" Archived 2023-07-20 at the Wayback Machine (On the solution of equations to which one is led by the investigation of the linear distribution of galvanic currents), Annalen der Physik und Chemie, 72 (12) : 497–508.

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