Bayesian phylogenetics aims at estimating phylogenetic trees together with evolutionary and population dynamic parameters based on genetic sequences. It has been noted that the clock rate, one of the evolutionary parameters, decreases with an increase in the sampling period of sequences. In particular, clock rates of epidemic outbreaks are often estimated to be higher compared with the long-term clock rate. Purifying selection has been suggested as a biological factor that contributes to this phenomenon, since it purges slightly deleterious mutations from a population over time. However, other factors such as methodological biases may also play a role and make a biological interpretation of results difficult. In this paper, we identify methodological biases originating from the choice of tree prior, that is, the model specifying epidemiological dynamics. With a simulation study we demonstrate that a misspecification of the tree prior can upwardly bias the inferred clock rate and that the interplay of the different models involved in the inference can be complex and nonintuitive. We also show that the choice of tree prior can influence the inference of clock rate on real-world Ebola virus (EBOV) datasets. While commonly used tree priors result in very high clock-rate estimates for sequences from the initial phase of the epidemic in Sierra Leone, tree priors allowing for population structure lead to estimates agreeing with the long-term rate for EBOV.