The research goal of this work is to investigate modeling patterns that recur in ontologies. Such patterns may originate from certain design solutions, and they may possibly indicate emerging ontology design patterns. We describe our tree-mining method for identifying the emerging design patterns. The method works in two steps: (1) we transform the ontology axioms in a tree shape in order to find axiom patterns; and then, (2) we use association analysis to mine co-occuring axiom patterns in order to extract emerging design patterns. We conduct an experimental study on a set of 331 ontologies from the BioPortal repository. We show that recurring axiom patterns appear across all individual ontologies, as well as across the whole set. In individual ontologies, we find frequent and non-trivial patterns with and without variables. Some of the former patterns have more than 300,000 occurrences. The longest pattern without a variable discovered from the whole ontology set has size 12, and it appears in 14 ontologies. To the best of our knowledge, this is the first method for automatic discovery of emerging design patterns in ontologies. Finally, we demonstrate that we are able to automatically detect patterns, for which we have manually confirmed that they are fragments of ontology design patterns described in the literature. Since our method is not specific to particular ontologies, we conclude that we should be able to discover new, emerging design patterns for arbitrary ontology sets.