Biologically Feasible Gene Trees, Reconciliation Maps and Informative Triples

The history of gene families - which are equivalent to \emph{event-labeled} gene trees - can be reconstructed from empirically estimated evolutionary event-relations containing pairs of orthologous, paralogous or xenologous genes. The question then arises as whether inferred event-labeled gene trees are \emph{biologically feasible}, that is, if there is a possible true history that would explain a given gene tree. In practice, this problem is boiled down to finding a reconciliation map - also known as DTL-scenario - between the event-labeled gene trees and a (possibly unknown) species tree. In this contribution, we first characterize whether there is a valid reconciliation map for binary event-labeled gene trees $T$ that contain speciation, duplication and horizontal gene transfer events and some unknown species tree $S$ in terms of "informative" triples that are displayed in $T$ and provide information of the topology of $S$. These informative triples are used to infer the unknown species tree $S$ for $T$. We obtain a similar result for non-binary gene trees. To this end, however, the reconciliation map needs to be further restricted. We provide a polynomial-time algorithm to decide whether there is a species tree for a given event-labeled gene tree, and in the positive case, to construct the species tree and the respective (restricted) reconciliation map. However, informative triples as well as DTL-scenarios have its limitations when they are used to explain the biological feasibility of gene trees. While reconciliation maps imply biological feasibility, we show that the converse is not true in general. Moreover, we show that informative triples do neither provide enough information to characterize "relaxed" DTL-scenarios nor non-restricted reconciliation maps for non-binary biologically feasible gene trees.