Table of contents |
2 Example 3 Facts 4 Types of Trees |
An undirected simple graph G is a tree if it satisfies one (and therefore all) of the following equivalent conditions:
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.
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 nn-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
Definitions
If G has finitely many vertices, say n of them, then the above statements are also equivalent to:
An undirected simple graph G is called a forest if it has no simple cycles.Example
Facts
Types of Trees
See also Tree structure, Tree data structure.