  In this paper, we investigate a conjecture by von Haeseler concerning the
Maximum Parsimony method for phylogenetic estimation, which was published by
the Newton Institute in Cambridge on a list of open phylogenetic problems in
2007. This conjecture deals with the question whether Maximum Parsimony trees
are hereditary. The conjecture suggests that a Maximum Parsimony tree for a
particular (DNA) alignment necessarily has subtrees of all possible sizes which
are most parsimonious for the corresponding subalignments. We answer the
conjecture affirmatively for binary alignments on five taxa but also show how
to construct examples for which Maximum Parsimony trees are not hereditary.
Apart from showing that a most parsimonious tree cannot generally be reduced to
a most parsimonious tree on fewer taxa, we also show that compatible most
parsimonious quartets do not have to provide a most parsimonious supertree.
Last, we show that our results can be generalized to Maximum Likelihood for
certain nucleotide substitution models.
