  The so-called binary perfect phylogeny with persistent characters has
recently been thoroughly studied in computational biology as it is less
restrictive than the well known binary perfect phylogeny. Here, we focus on the
notion of (binary) persistent characters, i.e. characters that can be realized
on a phylogenetic tree by at most one $0 \rightarrow 1$ transition followed by
at most one $1 \rightarrow 0$ transition in the tree, and analyze these
characters under different aspects. First, we illustrate the connection between
persistent characters and Maximum Parsimony, where we characterize persistent
characters in terms of the first phase of the famous Fitch algorithm.
Afterwards we focus on the number of persistent characters for a given
phylogenetic tree. We show that this number solely depends on the balance of
the tree. To be precise, we develop a formula for counting the number of
persistent characters for a given phylogenetic tree based on an index of tree
balance, namely the Sackin index. Lastly, we consider the question of how many
(carefully chosen) binary characters together with their persistence status are
needed to uniquely determine a phylogenetic tree and provide an upper bound for
the number of characters needed.
