  Applying a method to reconstruct a phylogenetic tree from random data
provides a way to detect whether that method has an inherent bias towards
certain tree `shapes'. For maximum parsimony, applied to a sequence of random
2-state data, each possible binary phylogenetic tree has exactly the same
distribution for its parsimony score. Despite this pleasing and slightly
surprising symmetry, some binary phylogenetic trees are more likely than others
to be a most parsimonious (MP) tree for a sequence of $k$ such characters, as
we show. For $k=2$, and unrooted binary trees on six taxa, any tree with a
caterpillar shape has a higher chance of being an MP tree than any tree with a
symmetric shape. On the other hand, if we take any two binary trees, on any
number of taxa, we prove that this bias between the two trees vanishes as the
number of characters grows. However, again there is a twist: MP trees on six
taxa are more likely to have certain shapes than a uniform distribution on
binary phylogenetic trees predicts, and this difference does not appear to
dissipate as $k$ grows.
