Ané–Eulenstein conjecture on maximal groves

Let $\mathcal{S}$ be a set of taxon sets, and let a grove be a collection of taxon sets with the potential to construct informative supertrees. A grove is maximal in $\mathcal{S}$ if it is not properly contained in another grove in $\mathcal{S}$. The following properties are equivalent: Ané–Eulenstein conjecture.

1. For any set $\mathcal{S}$ of taxon sets, the set of maximal groves in $\mathcal{S}$ is a partition of $\mathcal{S}$.
2. If two groves intersect, their union is a grove.
3. Two maximal groves do not intersect.

The conjecture concerns when overlap among taxon sets guarantees that collections of phylogenies can be combined informatively; the source attributes it to Ané et al. and presents it as a criterion relevant to identifying useful groves in databases. The paper later notes that the analogous property fails for informative groves in general.