Tree (graph theory)

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A labeled tree with 6 vertices and 5 edges

In graph theory, a tree is a graph in which any two vertices are connected by exactly one path. A forest is a graph in which any two vertices are connected by at most one path. An equivalent definition is that a forest is a disjoint union of trees (hence the name). Trees are widely used in Computer Science data structures such as binary search trees, heaps, tries, etc.

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

[edit] Definitions

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

G is connected and has no simple cycles.
G has no simple 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. 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. 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 n vertices are typically given the labels 1, 2, …, n.

A regular (or homogeneous) tree is a tree in which every vertex has the same degree. See regular graph.

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

[edit] 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.

[edit] Facts

Every tree is a bipartite 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.

[edit] 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

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