  Phylogenetic inference aims to reconstruct the evolutionary relationships of
different species based on genetic (or other) data. Discrete characters are a
particular type of data, which contain information on how the species should be
grouped together. However, it has long been known that some characters contain
more information than others. For instance, a character that assigns the same
state to each species groups all of them together and so provides no insight
into the relationships of the species considered. At the other extreme, a
character that assigns a different state to each species also conveys no
phylogenetic signal. In this manuscript, we study a natural combinatorial
measure of the information content of an individual character and analyse
properties of characters that provide the maximum phylogenetic information,
particularly, the number of states such a character uses and how the different
states have to be distributed among the species or taxa of the phylogenetic
tree.
