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Tree (graph theory)

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


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

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.


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

Types of Trees

See also Tree structure, Tree data structure.