'Bureaucratic' set systems, and their role in phylogenetics

We say that a collection $\text{Cc}$ of subsets of $X$ is {\em bureaucratic} if every maximal hierarchy on $X$ contained in $\text{Cc}$ is also maximum. We characterise bureaucratic set systems and show how they arise in phylogenetics. This framework has several useful algorithmic consequences: we generalize some earlier results and derive a polynomial-time algorithm for a parsimony problem arising in phylogenetic networks.