Modern problems of concept annotation associate an object of interest (gene, individual, text document) with a set of interrelated textual descriptors (functions, diseases, topics), often organized in concept hierarchies or ontologies. Most ontologies can be seen as directed acyclic graphs, where nodes represent concepts and edges represent relational ties between these concepts. Given an ontology graph, each object can only be annotated by a consistent subgraph; that is, a subgraph such that if an object is annotated by a particular concept, it must also be annotated by all other concepts that generalize it. Ontologies therefore provide a compact representation of a large space of possible consistent subgraphs; however, until now we have not been aware of a practical algorithm that can enumerate such annotation spaces for a given ontology. In this work we propose an algorithm for enumerating consistent subgraphs of directed acyclic graphs. The algorithm recursively partitions the graph into strictly smaller graphs until the resulting graph becomes a rooted tree (forest), for which a linear-time solution is computed. It then combines the tallies from graphs created in the recursion to obtain the final count. We prove the correctness of this algorithm and then apply it to characterize four major biomedical ontologies. We believe this work provides valuable insights into concept annotation spaces and predictability of ontological annotation.