Tree (graph theory)

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Trees
A labeled tree with 6 vertices and 5 edges
Vertices	v
Edges	v - 1
Chromatic number	2

In graph theory, a tree is a graph in which any two vertices are connected by exactly one path. Alternatively, any connected graph with no cycles is a tree. A forest is a disjoint union of trees. Trees are widely used in computer science data structures such as binary search trees, heaps, tries, Huffman trees for data compression, etc.

1 Definitions
2 Example
3 Facts
4 Enumeration
5 Types of trees
6 See also
7 References

Definitions

A tree is an undirected simple graph G that satisfies any of the following equivalent conditions:

G is connected and has no cycles.
G has no cycles, and a simple cycle is formed if any edge is added to G.
G is connected, and it is not connected anymore if any edge is removed from G.
G is connected and the 3-vertex complete graph $K 3$ is not a minor of G.
Any two vertices in G can be connected by a unique simple path.

If G has finitely many vertices, say n of them, then the above statements are also equivalent to any of the following conditions:

G is connected and has n − 1 edges.
G has no simple cycles and has n − 1 edges.

An undirected simple graph G is called a forest if it has no simple cycles.

A directed tree is a directed graph which would be a tree if the directions on the edges were ignored. Some authors restrict the phrase to the case where the edges are all directed towards a particular vertex, or all directed away from a particular vertex.

A tree is called a rooted tree if one vertex has been designated the root, in which case the edges have a natural orientation, towards or away from the root. The tree-order is the partial ordering on the vertices of a tree with u ≤ v if and only if the unique path from the root to v passes through u. A tree which is a subgraph of some graph G is a normal tree if the ends of every edge in G are comparable in this tree-order (Diestel 2005, p. 15). Rooted trees, often with additional structure such as ordering of the neighbors at each vertex, are a key data structure in computer science; see tree data structure. In a context where trees are supposed to have a root, a tree without any designated root is called a free tree.

A polytree has at most one undirected path between any two vertices. In other words, a polytree is a directed acyclic graph (DAG) for which there are no undirected cycles either.

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

An irreducible (or series-reduced) tree is a tree in which there is no vertex of degree 2.

An ordered tree is a tree for which an ordering is specified for the children of each node.

An n-ary tree is a tree for which each node which is not a leaf has at most n children. 2-ary trees (resp. 3-ary trees) are sometimes called binary trees (resp. ternary trees)

Example

The example tree shown to the right has 6 vertices and 6 − 1 = 5 edges. The unique simple path connecting the vertices 2 and 6 is 2-4-5-6.

Facts

Every tree is a bipartite graph and a median graph. Every tree with only countably many vertices is a planar graph.

Every connected graph G admits a spanning tree, which is a tree that contains every vertex of G and whose edges are edges of G. Every connected graph even admits a normal spanning tree (Diestel 2005, Prop. 1.5.6).

Every tree with $n \geq 2$ vertices has at least two leaves or vertices of degree 1. The minimal number of leaves corresponds to path graph but maximal number $(n - 1)$ corresponds to star graph.

For any three vertices in a tree, the three paths between them have at least one vertex in common.

Enumeration

Given n labeled vertices, there are nⁿ⁻² different ways to connect them to make a tree. This result is called Cayley's formula. It can be proved by first showing that the number of trees with n vertices of degree d₁,d₂,...,d_n is the multinomial coefficient

${n-2 \choose d_1-1, d_2-1, \ldots, d_n-1}.$

Counting the number of unlabeled trees is a harder problem. No closed formula for the number t(n) of trees with n vertices up to graph isomorphism is known. Otter (1948) proved that

$t(n) \sim C \alpha^n n^{-5/2} \quad\text{as } n\to\infty,$

with C = 0.53495… and α = 2.95576… (here, $f \sim g$ means that $\lim_{n \to \infty} f/g = 1$ ).

Types of trees

See List of graph theory topics: Trees.

References

Diestel, Reinhard (2005), Graph Theory (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-26183-4, <http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/> .
Otter, Richard (1948), "The Number of Trees", Annals of Mathematics, Second Series 49(3): 583–599, doi:10.2307/1969046 .