  Phylogenetic (i.e. leaf-labeled) trees play a fundamental role in
evolutionary research. A typical problem is to reconstruct such trees from data
like DNA alignments (whose columns are often referred to as characters), and a
simple optimization criterion for such reconstructions is maximum parsimony. It
is generally assumed that this criterion works well for data in which state
changes are rare. In the present manuscript, we prove that each phylogenetic
tree $T$ with $n\geq 20 k$ leaves is uniquely defined by the set $A_k(T)$,
which consists of all characters with parsimony score $k$ on $T$. This can be
considered as a promising first step towards showing that maximum parsimony as
a tree reconstruction criterion is justified when the number of changes in the
data is relatively small.
