# Improved Duplication-Transfer-Loss Reconciliation with
## 2 Definitions and Preliminaries
### DTL Reconciliation  
Given a rooted gene tree and a rooted species tree, Duplication-Transfer-Loss (DTL) reconciliation shows how the gene tree evolved within the species tree through speciation, gene duplication, transfer, and gene loss. Essentially, DTL reconciliation maps each node of the gene tree to a node or edge on the species tree and designates each internal node of the gene tree as representing either a speciation, duplication, or transfer event.  
Computed DTL reconciliations can sometimes be _time-inconsistent_ in that one or more inferred transfers may be inconsistent with any valid dating of the internal nodes of the species tree. The problem of finding an optimal _time-consistent_ DTL reconciliation is NP-hard [4; 24] and, consequently, DTL reconciliation algorithms often seek an optimal, not necessarily time-consistent, DTL reconciliation [4; 5; 7; 11; 13]. However, it is possible to perform additional filtering of computed optimal DTL reconciliations so that any time-inconsistent reconciliations are suppressed [8; 21]. Time-consistent DTL reconciliations can also be computed efficiently if the input species tree can be fully dated [3; 25]. In this work, we focus on computing optimal, not necessarily time-consistent, reconciliations with undated species trees, but the same techniques also extend to reconciliations with dated species trees.  
Almost all existing DTL reconciliation models have some implicit limitations in how they handle transfer events. For example, nearly all DTL reconciliation models implicitly assume that transfer events are _additive_, i.e., that the transferred gene inserts itself as a new gene in the recipient species instead of _replacing_ an existing homologous gene. Extensions of the DTL model that simultaneously model both additive and replacing transfers, referred to as the _DTLR reconciliation model_ have also been proposed [26; 27], but the problem of inferring optimal reconciliations under such models is known to be NP-hard [26; 27]. Likewise, almost all existing DTL reconciliation models, except for [10; 16], assume that transfer events only occur between branches, or species, represented on the given species tree. Thus, they often cannot correctly account for transfers from unsampled species lineages.  
In the following, we introduce an extended version of the DTL reconciliation model that can explicitly model two additional types of transfer events, as defined below. Our approach builds upon initial ideas first briefly mentioned (but not further developed, formalized, implemented, or tested), in a previous 2012 paper [7].  
**Definition 1** (Transfer-Loss (\(\mathbb{TL}\))).: _A transfer-loss (\(\mathbb{TL}\)) event occurs when the descendants of the donor species lose all copies derived from the transferred gene._  
Figure 1 provides an illustration of a \(\mathbb{TL}\) event and a detailed discussion occurs in Section 2.2 below. As noted earlier, \(\mathbb{TL}\) events were also explicitly defined and handled in [3; 16].  
**Definition 2** (Transfer from unsampled lineage (\(\mathbb{TX}\))).: _A transfer from unsampled lineage (\(\mathbb{TX}\)) event occurs when the donor species is not represented in the species tree._  
In this work, unsampled lineages are representative any lineage not present in the species tree, including those that are extinct. We refer to transfer events that are neither \(\mathbb{TL}\) events nor \(\mathbb{TX}\) events as \(\mathbb{T}\) events. Thus, \(\mathbb{T}\) represents the \"standard\" transfer event type that is modeled by default in existing DTL reconciliation models.  
Figure 1: Illustration of the difference between (**a**) a standard transfer event and (**b**) a \(\mathbb{TL}\) event.