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 b
inary 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.
In a recent paper, Klaere et al. modeled the impact of substitutions on arbitrary branches of a phylogenetic tree on an alignment site by the so-called One Step Mutation (OSM) matrix. By utilizing the concept of the OSM matrix for the four-state nucl
eotide alphabet, Nguyen et al. presented an efficient procedure to compute the minimal number of substitutions needed to translate one alignment site into another.The present paper delivers a proof for this computation.Moreover, we provide several mathematical insights into the generalization of the OSM matrix to multistate alphabets.The construction of the OSM matrix is only possible if the matrices representing the substitution types acting on the character states and the identity matrix form a commutative group with respect to matrix multiplication. We illustrate a means to establish such a group for the twenty-state amino acid alphabet and critically discuss its biological usefulness.
Tuffley and Steel (1997) proved that Maximum Likelihood and Maximum Parsimony methods in phylogenetics are equivalent for sequences of characters under a simple symmetric model of substitution with no common mechanism. This result has been widely cit
ed ever since. We show that small changes to the model assumptions suffice to make the two methods inequivalent. In particular, we analyze the case of bounded substitution probabilities as well as the molecular clock assumption. We show that in these cases, even under no common mechanism, Maximum Parsimony and Maximum Likelihood might make conflicting choices. We also show that if there is an upper bound on the substitution probabilities which is `sufficiently small, every Maximum Likelihood tree is also a Maximum Parsimony tree (but not vice versa).
In this paper we investigate mathematical questions concerning the reliability (reconstruction accuracy) of Fitchs maximum parsimony algorithm for reconstructing the ancestral state given a phylogenetic tree and a character. In particular, we conside
r the question whether the maximum parsimony method applied to a subset of taxa can reconstruct the ancestral state of the root more accurately than when applied to all taxa, and we give an example showing that this indeed is possible. A surprising feature of our example is that ignoring a taxon closer to the root improves the reliability of the method. On the other hand, in the case of the two-state symmetric substitution model, we answer affirmatively a conjecture of Li, Steel and Zhang which states that under a molecular clock the probability that the state at a single taxon is a correct guess of the ancestral state is a lower bound on the reconstruction accuracy of Fitchs method applied to all taxa.
