  The question whether a partition $\mathcal{P}$ and a hierarchy $\mathcal{H}$
or a tree-like split system $\mathfrak{S}$ are compatible naturally arises in a
wide range of classification problems. In the setting of phylogenetic trees,
one asks whether the sets of $\mathcal{P}$coincide with leaf sets of connected
components obtained by deleting some edges from the tree $T$ that represents
$\mathcal{H}$ or $\mathfrak{S}$, respectively. More generally, we ask whether a
refinement $T^*$ of $T$ exists such that $T^*$ and $\mathcal{P}$ are compatible
in this sense. The latter is closely related to the question as to whether
there exists a tree at all that is compatible with $\mathcal{P}$. We report
several characterizations for (refinements of) hierarchies and split systems
that are compatible with (systems of) partitions. In addition, we provide a
linear-time algorithm to check whether refinements of trees and a given
partition are compatible. The latter problem becomes NP-complete but
fixed-parameter tractable if a system of partitions is considered instead of a
single partition. In this context, we also explore the close relationship of
the concept of compatibility and so-called Fitch maps.
