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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).

## 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 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