Hello,

The answer to your first question is here :

http://mbe.oxfordjournals.org/content/17/6/839.full.pdf+html

Its a complex and interesting subject !

The Mk model in a maximum likelihood framework will give exactly the same tree as parsimony IF each of the characters in the analysis are allowed their own set of branchlengths for the tree (this was shown by Steel - the above paper is a good entry into this literature). If this is not the case, then the two methods need not coincide in choosing the same tree.  However, my guess is that they will tend to estimate the same tree if the 'rate' of character state change is pretty low in the tree of interest.  Parsimony will start to systematically estimate the wrong tree under 'long branch attraction' conditions as showed by Felsenstein in 1978.  In this case, you can expect Mk analyses with likelihood-based methods to differ from parsimony.  Regarding the question about a 'stationary distribution' for a phenotypic character....I guess it is roughly defined as the probabilities of the character states at the tips of the tree as the branchlengths go to infinity.  Or more precisely, the vector of probabilities of the character states occurirng at the end of a branch of a tree as the length of that branch goes to infinity.  In terms of its intuitive meaning...I guess it summarizes what you might expect to be the chance you'd see a given morphological character state after lots of changes in morphology have taken place over the tree.  I'm not sure it is very 'interpretable' in terms of morphological characters.

yes

This paper is also helpful in showing the relationship between the Mk model in Bayesian analysis and the parsimony model for morphological characters:

http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0109210

Vincent and Andrew, thank you so much for your answers! The reasons are now more or less clear to me, and the papers were very enlightening.

It is also the case that almost all the time, parsimony and model-based methods give the same or not significantly different results for molecular data.  See https://www.researchgate.net/publication/229912270_Do_model-based_phylogenetic_analyses_outperform_parsimony_A_test_with_empirical_data

Why this is so is not clear - perhaps the "long branch attraction" problem is not as pervasive as has been advertised.

Article Do model-based phylogenetic analyses outperform parsimony? A...

"Why this is so is not clear - perhaps the "long branch attraction" problem is not as pervasive as has been advertised."

That depends of course on the taxa and the genes you're looking at!

In morphology, the behavior of Bayesian inference (and maximum likelihood) has not been tested much and remains incompletely understood. In particular, the effects of realistic distributions of missing data are unknown. We do know what they do to parsimony: nothing... This and similar points are discussed in my "Reproducibility in phylogenetics" preprint.