Inferring gene duplications, transfers and losses can be done in a discrete framework
In the field of phylogenetics, the evolutionary history of a set of organisms is commonly depicted by a species tree—whose internal nodes represent speciation events—while the evolutionary history of a gene family is depicted by a gene tree—whose internal nodes can also represent macro-evolutionary events such as gene duplications and transfers. As speciation events are only part of the events shaping a gene history, the topology of a gene tree can show incongruences with that of the corresponding species tree. These incongruences can be used to infer the macro-evolutionary events undergone by the gene family. This is done by embedding the gene tree inside the species tree and hence providing a reconciliation of those trees. In the past decade, several parsimony-based methods have been developed to infer such reconciliations, accounting for gene duplications ($$\mathbb {D}$$D), transfers ($$\mathbb {T}$$T) and losses ($$\mathbb {L}$$L). The main contribution of this paper is to formally prove an important assumption implicitly made by previous works on these reconciliations, namely that solving the (maximum) parsimony $$\mathbb {DTL}$$DTL reconciliation problem in the discrete framework is equivalent to finding a most parsimonious $$\mathbb {DTL}$$DTL scenario in the continuous framework. In the process, we also prove several intermediate results that are useful on their own and constitute a theoretical toolbox that will likely facilitate future theoretical contributions in the field.