The most parsimonious tree for random data

• For random binary data, each binary tree has the same parsimony distribution.
• Yet not every tree is equally likely to be a most parsimonious (MP) tree.
• The bias of one given tree over another decays as the number of characters grows.
• But the bias of an MP tree towards certain shapes persists with more characters.

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=2k=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 k   grows. However, again there is a twist: MP trees on six taxa for k=2k=2 random binary characters are more likely to have certain shapes than a uniform distribution on binary phylogenetic trees predicts. Moreover, this shape bias appears, from simulations, to be more pronounced for larger values of k.

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