View source for Tree (graph theory)

{{Short description|Undirected, connected and acyclic graph}}
{{For|abstract data type|Tree (abstract data type)}}{{Distinguish|text=[[Trie]], a specific type of tree data structure}}
{{Infobox graph
 | name = Trees
 | image = [[File:Tree graph.svg|180px]]
 | image_caption = A labeled tree with 6 vertices and 5 edges.
 | vertices = ''v''
 | edges = ''v''&nbsp;−&nbsp;1
 | radius = 
 | diameter = 
 | girth = 
 | chromatic_number = 2 if ''v'' > 1
 | chromatic_index = 
 | properties = 
}}

In [[graph theory]], a '''tree''' is an [[undirected graph]] in which any two [[Vertex (graph theory)|vertices]] are connected by {{em|exactly one}} [[Path (graph theory)|path]], or equivalently a [[Connected graph|connected]] [[Cycle (graph theory)|acyclic]] undirected graph.{{sfn|Bender|Williamson|2010|p=171}} A '''forest''' is an undirected graph in which any two vertices are connected by {{em|at most one}} path, or equivalently an acyclic undirected graph, or equivalently a [[Disjoint union of graphs|disjoint union]] of trees.{{sfn|Bender|Williamson|2010|p=172}}

A directed tree,{{sfn|Deo|1974|p=206}} oriented tree,<ref name="Harary 1980">See {{harvtxt|Harary|Sumner|1980}}.</ref><ref name="s91">See {{harvtxt|Simion|1991}}.</ref> [[polytree]],<ref name="d99">See {{harvtxt|Dasgupta|1999}}.</ref> or singly connected network<ref name="kp83">See {{harvtxt|Kim|Pearl|1983}}.</ref> 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 [[Tree (data structure)|trees]] in [[computer science]] have [[underlying graph]]s that are trees in graph theory, although such data structures are generally [[Tree (graph theory)#Rooted tree|rooted trees]]. A rooted tree may be directed, called a directed rooted tree,<ref name="Williamson1985">{{cite book|author=Stanley Gill Williamson|title=Combinatorics for Computer Science|year=1985|publisher=Courier Dover Publications|isbn=978-0-486-42076-9|page=288}}</ref><ref name="multi">{{cite book|author1=Mehran Mesbahi|author2=Magnus Egerstedt|title=Graph Theoretic Methods in Multiagent Networks|year=2010|publisher=Princeton University Press|isbn=978-1-4008-3535-5|page=38}}</ref> either making all its edges point away from the root—in which case it is called an [[Arborescence (graph theory)|arborescence]]{{sfn|Deo|1974|p=206}}<ref name="DuKo2011">{{cite book|author1=Ding-Zhu Du|author2=Ker-I Ko|author3=Xiaodong Hu|title=Design and Analysis of Approximation Algorithms|year=2011|publisher=Springer Science & Business Media|isbn=978-1-4614-1701-9|page=108}}</ref> or out-tree{{sfn|Deo|1974|p=207}}<ref name="GrossYellen2013">{{cite book|author1=Jonathan L. Gross|author2=Jay Yellen|author3=Ping Zhang|author3-link=Ping Zhang (graph theorist)|title=Handbook of Graph Theory, Second Edition|year=2013|publisher=CRC Press|isbn=978-1-4398-8018-0|page=116}}</ref>—or making all its edges point towards the root—in which case it is called an anti-arborescence<ref name="KorteVygen2012b">{{cite book|author1=Bernhard Korte|authorlink1=Bernhard Korte|author2=Jens Vygen|title=Combinatorial Optimization: Theory and Algorithms|year=2012|publisher=Springer Science & Business Media|isbn=978-3-642-24488-9|page=28|edition=5th}}</ref> or in-tree.{{sfn|Deo|1974|p=207}}<ref name="MehlhornSanders2008">{{cite book|author1=Kurt Mehlhorn|author-link=Kurt Mehlhorn|author2=Peter Sanders|author2-link=Peter Sanders (computer scientist)|title=Algorithms and Data Structures: The Basic Toolbox|date=2008|publisher=Springer Science & Business Media|isbn=978-3-540-77978-0|pages=52 |url=http://people.mpi-inf.mpg.de/~mehlhorn/ftp/Toolbox/Introduction.pdf |archive-url=https://web.archive.org/web/20150908084811/http://people.mpi-inf.mpg.de/~mehlhorn/ftp/Toolbox/Introduction.pdf |archive-date=2015-09-08 |url-status=live}}</ref> A rooted tree itself has been defined by some authors as a directed graph.<ref name="Makinson2012">{{cite book|author=David Makinson|author-link=David Makinson|title=Sets, Logic and Maths for Computing|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4471-2499-3|pages=167–168}}</ref><ref>{{cite book|author=Kenneth Rosen|title=Discrete Mathematics and Its Applications, 7th edition|year=2011|publisher=McGraw-Hill Science|page=747|isbn=978-0-07-338309-5}}</ref><ref name="Schrijver2003">{{cite book|author=Alexander Schrijver|title=Combinatorial Optimization: Polyhedra and Efficiency|year=2003|publisher=Springer|isbn=3-540-44389-4|page=34}}</ref> 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 [[Arborescence (graph theory)|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 {{em|tree}} was coined in 1857 by the British mathematician [[Arthur Cayley]].<ref>Cayley (1857) [https://books.google.com/books?id=MlEEAAAAYAAJ&pg=PA172 "On the theory of the analytical forms called trees,"] ''Philosophical Magazine'', 4th series, '''13''' : 172–176.<br>However it should be mentioned that in 1847, [[Karl Georg Christian von Staudt|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 [https://books.google.com/books?id=MzQAAAAAQAAJ&pg=PA20 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) [https://books.google.com/books?id=gx4AAAAAMAAJ&q=Kirchoff&pg=PA497 "Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird"] {{Webarchive|url=https://web.archive.org/web/20230720150251/https://books.google.com/books?id=gx4AAAAAMAAJ&q=Kirchoff&pg=PA497 |date=2023-07-20 }} (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.</ref>

==Definitions==

===Tree===
A ''tree'' is an undirected graph {{mvar|G}} that satisfies any of the following equivalent conditions:
* {{mvar|G}} is [[Connected graph|connected]] and [[Cycle (graph theory)|acyclic]] (contains no cycles).
* {{mvar|G}} is acyclic, and a simple cycle is formed if any [[Edge (graph theory)|edge]] is added to {{mvar|G}}.
* {{mvar|G}} is connected, but would become [[Connectivity (graph theory)#Connected graph|disconnected]] if any single edge is removed from {{mvar|G}}.
* {{mvar|G}} is connected and the [[complete graph]] {{math|''K''{{sub|3}}}} is not a [[Minor (graph theory)|minor]] of {{mvar|G}}.
* Any two vertices in {{mvar|G}} can be connected by a unique [[Path (graph theory)|simple path]].
If {{mvar|G}} has finitely many vertices, say {{mvar|n}} of them, then the above statements are also equivalent to any of the following conditions:
* {{mvar|G}} is connected and has {{math|''n'' − 1}} edges.
* {{mvar|G}} is connected, and every [[subgraph (graph theory)|subgraph]] of {{mvar|G}} includes at least one vertex with zero or one incident edges. (That is, {{mvar|G}} is connected and [[Degeneracy (graph theory)|1-degenerate]].)
* {{mvar|G}} has no simple cycles and has {{math|''n'' − 1}} edges.
As elsewhere in graph theory, the [[order-zero graph]] (graph with no vertices) is generally not considered to be a tree: while it is vacuously connected as a graph (any two vertices can be connected by a path), it is not [[n-connected|0-connected]] (or even (−1)-connected) in algebraic topology, unlike non-empty trees, and violates the "one more vertex than edges" relation. It may, however, be considered as a forest consisting of zero trees.

An {{em|internal vertex}} (or inner vertex) is a vertex of [[Degree (graph theory)|degree]] at least 2. Similarly, an {{em|external vertex}} (or outer vertex, terminal vertex or leaf) is a vertex of degree 1. A branch vertex in a tree is a vertex of degree at least 3.<ref>{{cite arXiv |last1=DeBiasio |first1=Louis |last2=Lo |first2=Allan |date=2019-10-09 |title=Spanning trees with few branch vertices |class=math.CO |eprint=1709.04937 }}</ref>

An {{em|irreducible tree}} (or series-reduced tree) is a tree in which there is no vertex of degree 2 (enumerated at sequence {{OEIS link|A000014}} in the [[OEIS]]).{{sfn|Harary|Prins|1959|p=150}}

===Forest===
A {{em|forest}} is an undirected acyclic graph or equivalently a [[disjoint union of graphs|disjoint union]] of trees.  Trivially so, each connected component of a forest is a tree. As special cases, the order-zero graph (a forest consisting of zero trees), a single tree, and an edgeless graph, are examples of forests.
Since for every tree {{math|1=''V'' − ''E'' = 1}}, we can easily count the number of trees that are within a forest by subtracting the difference between total vertices and total edges. {{math|1=''V'' − ''E'' =}} number of trees in a forest.

===Polytree===
{{main|Polytree}}
A {{em|polytree}}<ref name="d99"/> (or ''directed tree''{{sfn|Deo|1974|p=206}} or ''oriented tree''<ref name="Harary 1980"/><ref name="s91"/> or ''singly connected network''<ref name="kp83"/>) is a [[directed acyclic graph]] (DAG) whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic.

Some authors restrict the phrase "directed tree" to the case where the edges are all directed towards a particular vertex, or all directed away from a particular vertex (see [[Arborescence (graph theory)|arborescence]]).<ref>{{cite journal
 | last = Chen | first = Wai-kai
 | doi = 10.1137/0114048
 | journal = [[SIAM Journal on Applied Mathematics]]
 | mr = 209064
 | pages = 550–560
 | title = On directed trees and directed {{mvar|k}}-trees of a digraph and their generation
 | volume = 14
 | year = 1966| issue = 3
 }}</ref><ref name=kozlov>{{cite journal
 | last = Kozlov | first = Dmitry N.
 | doi = 10.1006/jcta.1999.2984
 | issue = 1
 | journal = Journal of Combinatorial Theory
 | mr = 1713484
 | pages = 112–122
 | series = Series A
 | title = Complexes of directed trees
 | volume = 88
 | year = 1999}}</ref><ref>{{citation
 | last1 = Tran | first1 = Ngoc Mai
 | last2 = Buck | first2 = Johannes
 | last3 = Klüppelberg | first3 = Claudia
 | arxiv = 2102.06197
 | date = February 2024
 | doi = 10.1093/jrsssb/qkad165
 | journal = Journal of the Royal Statistical Society Series B: Statistical Methodology
 | title = Estimating a directed tree for extremes| volume = 86
 | issue = 3
 | pages = 771–792
 }}</ref>

===Polyforest===
A {{em|polyforest}} (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is acyclic.

As with directed trees, some authors restrict the phrase "directed forest" to the case where the edges of each connected component are all directed towards a particular vertex, or all directed away from a particular vertex (see [[Arborescence (graph theory)|branching]]).<ref name=kozlov/>

===Rooted tree===
A {{em|rooted tree}} is a tree in which one vertex has been designated the root.{{sfn|Bender|Williamson|2010|p=173}} The edges of a rooted tree can be assigned a natural orientation, either away from or towards the root, in which case the structure becomes a directed rooted tree. When a directed rooted tree has an orientation away from the root, it is called an ''arborescence''{{sfn|Deo|1974|p=206}} or ''out-tree'';{{sfn|Deo|1974|p=207}} when it has an orientation towards the root, it is called an ''anti-arborescence'' or ''in-tree''.{{sfn|Deo|1974|p=207}} The tree-order is the [[partial ordering]] on the vertices of a tree with {{math|''u'' < ''v''}} if and only if the unique path from the root to {{mvar|v}} passes through {{mvar|u}}. A rooted tree {{mvar|T}} that is a [[Glossary of graph theory#Subgraphs|subgraph]] of some graph {{mvar|G}} is a [[normal tree]] if the ends of every {{mvar|T}}-path in {{mvar|G}} are comparable in this tree-order {{harv|Diestel|2005|p=15}}. Rooted trees, often with an additional structure such as an ordering of the neighbors at each vertex, are a key data structure in computer science; see [[tree data structure]].

In a context where trees typically have a root, a tree without any designated root is called a ''free tree''.

A ''labeled tree'' is a tree in which each vertex is given a unique label. The vertices of a labeled tree on {{mvar|n}} vertices (for nonnegative integers {{mvar|n}}) are typically given the labels {{math|1, 2, …, ''n''}}. A ''[[recursive tree]]'' is a labeled rooted tree where the vertex labels respect the tree order (i.e., if {{math|''u'' < ''v''}} for two vertices {{mvar|u}} and {{mvar|v}}, then the label of {{mvar|u}} is smaller than the label of {{mvar|v}}).

In a rooted tree, the ''parent'' of a vertex {{mvar|v}} is the vertex connected to {{mvar|v}} on the [[Path (graph theory)|path]] to the root; every vertex has a unique parent, except the root has no parent.{{sfn|Bender|Williamson|2010|p=173}} A ''child'' of a vertex {{mvar|v}} is a vertex of which {{mvar|v}} is the parent.{{sfn|Bender|Williamson|2010|p=173}} An ''ascendant'' of a vertex {{mvar|v}} is any vertex that is either the parent of {{mvar|v}} or is (recursively) an ascendant of a parent of {{mvar|v}}. A ''descendant'' of a vertex {{mvar|v}} is any vertex that is either a child of {{mvar|v}} or is (recursively) a descendant of a child of {{mvar|v}}. A ''sibling'' to a vertex {{mvar|v}} is any other vertex on the tree that shares a parent with {{mvar|v}}.{{sfn|Bender|Williamson|2010|p=173}} A ''leaf'' is a vertex with no children.{{sfn|Bender|Williamson|2010|p=173}} An ''internal vertex'' is a vertex that is not a leaf.{{sfn|Bender|Williamson|2010|p=173}}

The ''height'' of a vertex in a rooted tree is the length of the longest downward path to a leaf from that vertex. The ''height'' of the tree is the height of the root. The ''depth'' of a vertex is the length of the path to its root (''root path''). The depth of a tree is the maximum depth of any vertex. Depth is commonly needed in the manipulation of the various self-balancing trees, [[AVL tree]]s in particular. The root has depth zero, leaves have height zero, and a tree with only a single vertex (hence both a root and leaf) has depth and height zero. Conventionally, an empty tree (a tree with no vertices, if such are allowed) has depth and height −1.

A ''[[k-ary tree|{{mvar|k}}-ary tree]]'' (for nonnegative integers {{mvar|k}}) is a rooted tree in which each vertex has at most {{mvar|k}} children.<ref>See {{cite web|url=http://xlinux.nist.gov/dads/HTML/karyTree.html|title=k-ary tree|last=Black|first=Paul E.|date=4 May 2007|publisher=U.S. National Institute of Standards and Technology|access-date=8 February 2015|archive-date=8 February 2015|archive-url=https://web.archive.org/web/20150208124845/http://xlinux.nist.gov/dads/HTML/karyTree.html|url-status=live}}</ref> 2-ary trees are often called ''[[binary tree]]s'', while 3-ary trees are sometimes called ''[[ternary tree]]s''.

=== {{anchor|Plane tree}} Ordered tree===
An ''ordered tree'' (alternatively, ''plane tree'' or ''positional tree''<ref>{{cite book |last1=Cormen |first1=Thomas H. |last2=Leiserson |first2=Charles E. |last3=Rivest |first3=Ronald L. |last4=Stein |first4=Clifford |title=Introduction to Algorithms |date=2022 |publisher=MIT Press |location=Section B.5.3, ''Binary and positional trees'' |isbn=9780262046305 |page=1174 |edition=4th |url=https://mitpress.mit.edu/9780262046305/introduction-to-algorithms/ |access-date=20 July 2023 |archive-date=16 July 2023 |archive-url=https://web.archive.org/web/20230716082232/https://mitpress.mit.edu/9780262046305/introduction-to-algorithms/ |url-status=live }}</ref>) is a rooted tree in which an ordering is specified for the children of each vertex.{{sfn|Bender|Williamson|2010|p=173}}<ref>{{citation|title=Enumerative Combinatorics, Vol. I|volume=49|series=Cambridge Studies in Advanced Mathematics|first=Richard P.|last=Stanley|author-link=Richard P. Stanley|publisher=Cambridge University Press|year=2012|isbn=9781107015425|page=573|url=https://books.google.com/books?id=0wmJntp8IBQC&pg=PA573}}</ref> This is called a "plane tree" because an ordering of the children is equivalent to an embedding of the tree in the plane, with the root at the top and the children of each vertex lower than that vertex. Given an embedding of a rooted tree in the plane, if one fixes a direction of children, say left to right, then an embedding gives an ordering of the children. Conversely, given an ordered tree, and conventionally drawing the root at the top, then the child vertices in an ordered tree can be drawn left-to-right, yielding an essentially unique planar embedding.

==Properties==
* Every tree is a [[bipartite graph]]. A graph is bipartite if and only if it contains no cycles of odd length. Since a tree contains no cycles at all, it is bipartite.
* Every tree with only [[Countable set|countably]] many vertices is a [[planar graph]].
* Every connected graph ''G'' admits a [[spanning tree (mathematics)|spanning tree]], which is a tree that contains every vertex of ''G'' and whose edges are edges of ''G''. More specific types spanning trees, existing in every connected finite graph, include [[depth-first search]] trees and [[breadth-first search]] trees. Generalizing the existence of depth-first-search trees, every connected graph with only [[Countable set|countably]] many vertices has a [[Trémaux tree]].{{sfnp|Diestel|2005|loc=Prop. 8.2.4}} However, some [[Uncountable set|uncountable]]-[[Order of a graph|order]] graphs do not have such a tree.{{sfnp|Diestel|2005|loc=Prop. 8.5.2}}
* Every finite tree with ''n'' vertices, with {{nowrap|''n'' > 1}}, has at least two terminal vertices (leaves). This minimal number of leaves is characteristic of [[path graph]]s; the maximal number, {{nowrap|''n'' − 1}}, is attained only by [[star graph]]s. The number of leaves is at least the maximum vertex degree.
* For any three vertices in a tree, the three paths between them have exactly one vertex in common. More generally, a vertex in a graph that belongs to three shortest paths among three vertices is called a median of these vertices. Because every three vertices in a tree have a unique median, every tree is a [[median graph]].
* Every tree has a [[graph center|center]] consisting of one vertex or two adjacent vertices. The center is the middle vertex or middle two vertices in every longest path. Similarly, every ''n''-vertex tree has a centroid consisting of one vertex or two adjacent vertices. In the first case removal of the vertex splits the tree into subtrees of fewer than ''n''/2 vertices. In the second case, removal of the edge between the two centroidal vertices splits the tree into two subtrees of exactly ''n''/2 vertices.
* The maximal cliques of a tree are precisely its edges, implying that the class of trees has [[Graphs with few cliques|few cliques]].

==Enumeration==

=== Labeled trees ===
[[Cayley's formula]] states that there are {{math|''n''{{sup|''n''−2}}}} trees on {{mvar|n}} labeled vertices. A classic proof uses [[Prüfer sequence]]s, which naturally show a stronger result: the number of trees with vertices {{math|1, 2, …, ''n''}} of degrees {{math|''d''{{sub|1}}, ''d''{{sub|2}}, …, ''d{{sub|n}}''}} respectively, is the [[Multinomial theorem|multinomial coefficient]]
: <math>{n - 2 \choose d_1 - 1, d_2 - 1, \ldots, d_n - 1}.</math>

A more general problem is to count [[spanning tree]]s in an [[undirected graph]], which is addressed by the [[matrix tree theorem]]. (Cayley's formula is the special case of spanning trees in a [[complete graph]].) The similar problem of counting all the subtrees regardless of size is [[Sharp-P-complete|#P-complete]] in the general case ({{harvtxt|Jerrum|1994}}).

===Unlabeled trees===
Counting the number of unlabeled free trees is a harder problem. No closed formula for the number {{math|''t''(''n'')}} of trees with {{mvar|n}} vertices [[up to]] [[graph isomorphism]] is known. The first few values of {{math|''t''(''n'')}} are
: 1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551, 1301, 3159, … {{OEIS|A000055}}.
{{harvtxt|Otter|1948}} proved the asymptotic estimate
: <math>t(n) \sim C \alpha^n n^{-5/2} \quad\text{as } n\to\infty,</math>
with {{math|''C'' ≈ 0.534949606...}} and {{math|''α'' ≈ 2.95576528565...}} {{OEIS|A051491}}. Here, the {{math|~}} symbol means that 
:<math>\lim_{n \to \infty} \frac{t(n)}{C \alpha^n n^{-5/2} } = 1.</math>
This is a consequence of his asymptotic estimate for the number {{math|''r''(''n'')}} of unlabeled rooted trees with {{mvar|n}} vertices:
: <math>r(n) \sim D\alpha^n n^{-3/2} \quad\text{as } n\to\infty,</math>
with {{math|D ≈ 0.43992401257...}} and the same {{mvar|α}} as above (cf. {{harvtxt|Knuth|1997}}, chap. 2.3.4.4 and {{harvtxt|Flajolet|Sedgewick|2009}}, chap. VII.5, p.&nbsp;475).

The first few values of {{math|''r''(''n'')}} are<ref>See {{harvtxt|Li|1996}}.</ref>
: 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, 32973, … {{OEIS|A000081}}.

==Types of trees==
* A ''[[path graph]]'' (or ''linear graph'') consists of {{mvar|n}} vertices arranged in a line, so that vertices {{mvar|i}} and {{math|''i'' + 1}} are connected by an edge for {{math|1=''i'' = 1, …, ''n'' – 1}}.
* A ''[[starlike tree]]'' consists of a central vertex called ''root'' and several path graphs attached to it. More formally, a tree is starlike if it has exactly one vertex of degree greater than 2.
* A ''[[Star (graph theory)|star tree]]'' is a tree which consists of a single internal vertex (and {{math|''n'' – 1}} leaves). In other words, a star tree of order {{mvar|n}} is a tree of order {{mvar|n}} with as many leaves as possible.
* A ''[[caterpillar tree]]'' is a tree in which all vertices are within distance 1 of a central path subgraph.
* A ''[[List of graphs#Lobster|lobster tree]]'' is a tree in which all vertices are within distance 2 of a central path subgraph.
* A ''regular tree'' of degree {{mvar|d}} is the infinite tree with {{mvar|d}} edges at each vertex. These arise as the [[Cayley graph]]s of [[free group]]s, and in the theory of [[Building (mathematics)|Tits buildings]]. In [[statistical mechanics]] they are known as ''[[Bethe lattice]]s''.

==See also==
* [[Decision tree]]
* [[Hypertree]]
* [[Multitree]]
* [[Pseudoforest]]
* [[Tree structure]] (general)
* [[Tree (data structure)]]
* [[Unrooted binary tree]]

==Notes==
{{Reflist|30em}}

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==Further reading==
{{Commons category|Tree (graph theory)}}
* {{Citation | last1=Diestel | first1=Reinhard | author-link = Reinhard Diestel | title=Graph Theory | url=http://diestel-graph-theory.com/index.html | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | isbn=978-3-540-26183-4 | year=2005 }}.
* {{citation | last1=Flajolet | first1= Philippe |last2=Sedgewick|first2=Robert| author-link2=Robert Sedgewick (computer scientist)| title=Analytic Combinatorics|title-link= Analytic Combinatorics | isbn=978-0-521-89806-5|year=2009|publisher=Cambridge University Press}}
* {{springer|title=Tree|id=p/t094060|ref=none}}
* {{citation | last=Knuth | first=Donald E. | author-link=Donald E. Knuth | title=The Art of Computer Programming Volume 1: Fundamental Algorithms | date=November 14, 1997 | publisher=Addison-Wesley Professional | edition=3rd}}
* {{citation | doi=10.1016/0020-0190(94)00085-9 | last=Jerrum | first = Mark | year=1994 | title=Counting trees in a graph is #P-complete | journal=Information Processing Letters | volume=51 | issue=3 | pages=111–116 | issn=0020-0190 }}.
* {{citation | first1=Richard | last1=Otter | year=1948 | title=The Number of Trees | journal=Annals of Mathematics |series=Second Series | volume=49 | issue=3 | pages=583–599 | doi=10.2307/1969046 | jstor=1969046 }}.

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[[Category:Trees (graph theory)|*]]
[[Category:Bipartite graphs]]