  Reverse search is a convenient method for enumerating structured objects,
that can be used both to address theoretical issues and to solve data mining
problems. This method has already been successfully developed to handle
unordered trees. If the literature proposes solutions to enumerate singletons
of trees, we study in this article a more general problem, the enumeration of
sets of trees -- forests. Specifically, we mainly study irredundant forests,
i.e., where no tree is a subtree of another. By compressing each such forest
into a Directed Acyclic Graph (DAG), we develop a reverse search like method to
enumerate DAGs compressing irredundant forests. Remarkably, we prove that these
DAGs are in bijection with the row-Fishburn matrices, a well-studied class of
combinatorial objects. In a second step, we derive our irredundant forest
enumeration to provide algorithms for tackling related problems: (i)
enumeration of forests in their classical sense (where redundancy is allowed);
(ii) the enumeration of "subforests" of a forest, and (iii) the frequent
"subforest" mining problem. All the methods presented in this article enumerate
each item uniquely, up to isomorphism.
