  Most phylogenetic models assume that the evolutionary process is stationary
and reversible. As a result, the root of the tree cannot be inferred as part of
the analysis because the likelihood of the data does not depend on the position
of the root. Yet defining the root of a phylogenetic tree is a key component of
phylogenetic inference because it provides a point of reference for polarising
ancestor/descendant relationships and therefore interpreting the tree. In this
paper we investigate the effect of relaxing the reversibility assumption and
allowing the position of the root to be another unknown in the model. We
propose two hierarchical models that are centred on a reversible model but
perturbed to allow non-reversibility. The models differ in the degree of
structure imposed on the perturbations. The analysis is performed in the
Bayesian framework using Markov chain Monte Carlo methods. We illustrate the
performance of the two non-reversible models in analyses of simulated data sets
using two types of topological priors. We then apply the models to a real
biological data set, the radiation of polyploid yeasts, for which there is a
robust biological opinion about the root position. Finally we apply the models
to a second biological data set for which the rooted tree is controversial: the
ribosomal tree of life. We compare the two non-reversible models and conclude
that both are useful in inferring the position of the root from real biological
data sets.
