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One of the main aims of phylogenetics is the reconstruction of the correctevolutionary tree when data concerning the underlying species set are given.These data typically come in the form of DNA, RNA or protein alignments, whichconsist of various characters (also often referred to as sites). Often,however, tree reconstruction methods based on criteria like maximum parsimonymay 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 thatgenerated the data). On the other hand it has long been known that if thealignment consists of all the characters that correspond to edges of aparticular tree, i.e. they all require exactly $k=1$ substitution to berealized on that tree, then this tree will be recovered by maximum parsimonymethods. This is based on Buneman's theorem in mathematical phylogenetics. Itis 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 uniquemaximum parsimony tree (and we also show that this can be generalized tocharacters which require at most $k=2$ substitutions). In particular, this alsoproves a conjecture based on a recently published observation by Goloboff etal. affirmatively for the special case of $k=2$.

On the uniqueness of the maximum parsimony tree for data with up to two substitutions: an extension of the classic Buneman theorem in phylogenetics