Heuristics for unrooted, unranked, and ranked anomaly zones under birth-death models

Species trees that can generate a nonmatching gene tree topology that is more probable than the topology matching the species tree are said to be in an anomaly zone. We introduce some heuristic approaches to infer whether species trees are in anomaly zones when it is difficult or impossible to compute the entire distribution of gene tree topologies. Here, probabilities of unrooted, unranked, and ranked gene tree topologies under the multispecies coalescent are used. A ranked tree can be viewed as an unranked tree with a temporal ordering of its internal nodes. Overall, considering probabilities of unrooted or unranked gene tree topologies within one nearest neighbor interchange from the species tree topology is a reasonable heuristic to infer the existence of anomalous unrooted or unranked gene trees, respectively. We investigated a test proposed by Linkem et al. (2016) which classifies a species tree as being in an unranked anomaly zone if there is a subset of four taxa in an unranked anomaly zone. We find this test to have high true positive rates, but it can also have high false positive rates. For ranked trees, because at least one of the most probable ranked gene tree topologies must have the same unranked topology as the species tree, we propose to use only those ranked gene trees that have topologies that match the unranked species tree topology. We find that the probability that the species tree is in unrooted and unranked anomaly zones tends to increase with the speciation rate, and the probability of all three types of anomaly zones increases rapidly with the number of taxa. We find that probabilities that species trees are in an anomaly zone can be quite high for moderately high speciation rates.