The inference of new information on the relatedness of species by phylogenetic trees based on DNA data is one of the main challenges of modern biology. But despite all technological advances, DNA sequencing is still a time-consuming and costly process. Therefore, decision criteria would be desirable to decide a priori which data might contribute new information to the supertree which is not explicitly displayed by any input tree. A new concept, so-called groves, to identify taxon sets with the potential to construct such informative supertrees was suggested by Ané et al. in 2009. But the important conjecture that maximal groves can easily be identified in a database remained unproved and was published on the Isaac Newton Institute's list of open phylogenetic problems. In this paper, we show that the conjecture does not generally hold, but also introduce a new concept, namely 2-overlap groves, which overcomes this problem.