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.