  One of the main aims of phylogenetics is the reconstruction of the correct
evolutionary tree when data concerning the underlying species set are given.
These data typically come in the form of DNA, RNA or protein alignments, which
consist of various characters (also often referred to as sites). Often,
however, tree reconstruction methods based on criteria like maximum parsimony
may fail to provide a unique tree for a given dataset, or, even worse,
reconstruct the `wrong' tree (i.e. a tree that differs from the one that
generated the data). On the other hand it has long been known that if the
alignment consists of all the characters that correspond to edges of a
particular tree, i.e. they all require exactly $k=1$ substitution to be
realized on that tree, then this tree will be recovered by maximum parsimony
methods. This is based on Buneman's theorem in mathematical phylogenetics. It
is the goal of the present manuscript to extend this classic result as follows:
We prove that if an alignment consists of all characters that require exactly
$k=2$ substitutions on a particular tree, this tree will always be the unique
maximum parsimony tree (and we also show that this can be generalized to
characters which require at most $k=2$ substitutions). In particular, this also
proves a conjecture based on a recently published observation by Goloboff et
al. affirmatively for the special case of $k=2$.
