A forest is an acyclic graph (i.e., a graph without any graph cycles). Forests therefore consist only of (possibly disconnected) trees, hence the name "forest."
Examples of forests include the singleton graph, empty graphs, and all trees.
A forest with 
components and 
nodes has 
graph edges.
The numbers of forests on 
,
2, ... nodes are 1, 2, 3, 6, 10, 20, 37, ... (OEIS A005195).
A graph can be tested to determine if it is acyclic (i.e., a forest) in the Wolfram Language using AcylicGraphQ[g]. A collection of acyclic graphs is available as GraphData["Acyclic"] or GraphData["Forest"].
The total numbers of trees in all the forests of orders 
, 2, ... are 1, 3, 6, 13, 24, 49, 93, 190, 381, ... (OEIS
A005196). The average numbers of trees are
therefore 1, 3/2, 2, 13/6, 12/5, 49/20, 93/37, 5/2, ... (OEIS A095131
and A095132).
The triangle of numbers of 
-node
forests containing 
trees is 1; 1, 1; 1, 1, 1; 2, 2, 1, 1; 3, 3, 2, 1, 1; ... (OEIS A095133).
