  In this paper, we express the reduction types of Picard curves in terms of
tropical invariants associated to binary quintics. These invariants are
connected to Picard modular forms using recent work by Cl{\'e}ry and van der
Geer. We furthermore give a general framework for tropical invariants
associated to group actions on arbitrary varieties. The previous problem fits
in this general framework by mapping the space of binary forms to symmetrized
versions of the Deligne--Mumford compactification $\bar{M}_{0,n}$. We
conjecture that the techniques introduced here can be used to find tropical
invariants for binary forms of any degree.
