Tree (graph theory)

A 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).

Table of contents

1 Definitions
2 Example
3 Facts
4 Types of Trees

Definitions

An undirected simple graph G is a tree if it satisfies one (and therefore all) of the following equivalent conditions:

G is connected and has no simple cycles
G has no simple cycles and, if any edge is added to G, then a simple cycle is formed
G is connected and, if any edge is removed from G, then it is not connected anymore
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:

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.

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 planar and bipartite.

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.

Given n different vertices, there are n^n-2 different ways to connect them to make a tree. No closed formula for the number t(n) of trees with n vertices up to graph isomorphism is known. However, the asymptotic behavior of t(n) is known: there are numbers α≈3 and β≈0.5 such that

Types of Trees

Free tree
Rooted tree
- Ordered tree
Binary tree
- Full binary tree
Positional tree
Empty tree
K-ary tree
Charles' tree