# Improved Duplication-Transfer-Loss Reconciliation with
## 2 Definitions and Preliminaries
### DTLx Reconciliation
_If_ \(g\in I(G^{\prime})\setminus I(G)\) _and_ \(g^{\prime}\) _denotes it's unique child, then,_ 1. \(g\in\Theta_{\mathbb{L}}\cup\Theta_{\mathcal{X}}\)_, and_ \((g,g^{\prime})\in\Xi\)__ 2. \(\mathcal{M}(g)\) _and_ \(\tau(g)\) _are incomparable, and_ \(\mathcal{M}(g^{\prime})\leq_{S^{\prime}}\tau(g)\)__ 3. \(g\in\Theta_{\mathbb{X}}\) _if and only if_ \(\mathcal{M}(g)\in\mathcal{X}(S^{\prime})\)_._  
Constraint 1 specifies how the gene tree can be augmented with hidden nodes so that any inferred \(\mathbb{TL}\) and \(\mathbb{TX}\) events can be shown in the final reconciliation. Constraint 2 ensures that the mapping \(\mathcal{M}\) is consistent with the leaf-mapping \(\mathcal{L}\). Because \(Le(G^{\prime})\) is equivalent to \(Le(G)\), the leaf mappings of \(G^{\prime}\) into \(S^{\prime}\) must be equivalent to those of \(G\) into \(S\). Constraints 3a and 4 impose on \(\mathcal{M}\) the temporal constraints implied by \(S^{\prime}\). Constraint 3b implies that any non-hidden internal node in \(G^{\prime}\) may represent at most one transfer event. Constraint 5 determines the edges of \(G^{\prime}\) that are \(\mathbb{T}\), \(\mathbb{TL}\), or \(\mathbb{TX}\) edges. Constraints 6a-c state the conditions under which a non-hidden internal node of \(G^{\prime}\) may represent a speciation, duplication, and transfer, respectively. Constraint 6d specifies which species may be designated as the recipient species for any given \(\mathbb{T}\) event. Constraint 7a ensures that only those hidden nodes that represent a valid \(\mathbb{TL}\) or \(\mathbb{TX}\) event can appear in \(G^{\prime}\). Constraint 7b specifies which species may be designated as the recipient species for any given \(\mathbb{TL}\) or \(\mathbb{TX}\) event. Finally, Constraint 7c differentiates \(\mathbb{TL}\) from \(\mathbb{TX}\) events and requires that any \(\mathbb{TX}\) event must originate along an extra edge (representing unsampled lineages) of the augmented species tree. Figure 4 shows examples of DTLx reconciliations and illustrates how allowing for \(\mathbb{TL}\) and \(\mathbb{TX}\) events can lead to improved reconciliations. Given a DTLx reconciliation \(\alpha\), one can directly count the minimum number of gene losses implied by \(\alpha\) similarly as under the DTL reconciliation model [7], but with two additional considerations: First, any losses associated with \(\mathbb{TL}\) and \(\mathbb{TX}\) events (i.e., associated with hidden nodes of \(G^{\prime}\)) must be appropriately counted. Furthermore, second, losses must be counted using the original, unaugmented species tree \(S\), rather than using \(S^{\prime}\), except in cases where a gene tree node maps to an extra leaf. Specifically, to count the minimum number of losses that occur along an edge \((g,g^{\prime})\) of the gene tree, we must count the number of non-extra species lineages in the species tree that branch off from the path \(\mathcal{M}(g)\rightarrow_{S^{\prime}}\mathcal{M}(g^{\prime})\), and also account for any extra edges along the path. To facilitate this counting, we define the loss-distance between two nodes \(x,y\in V(S^{\prime})\), where \(y\leq_{S^{\prime}}x\), denoted \(d_{S^{\prime}}(x,y)\), as follows:\[d_{S^{\prime}}(x,y)= \begin{cases}0,&\text{if }x=y,\\ |\{e\colon e\in E(S)\text{ and }e\text{ fully or partially overlaps with the path }x\to_{S^{\prime}}y\}|,&\text{if }y\
ot\in\mathcal{X}(S^{\prime})\\ |\{e\colon e\in E(S)\text{ and }e\text{ fully or partially overlaps with the path }x\to_{S^{\prime}}y\}|+1,&\text{otherwise}\end{cases} \]  
For example, in the species tree \(S^{\prime}\) shown in Figure 4f, \(d_{S^{\prime}}(n_{2},B)=4\) and \(d_{S^{\prime}}(n_{2},e_{2})=2\). Figure 4: Impact of \(\mathbb{TL}\) and \(\mathbb{TX}\) events on optimal DTLx reconciliations. (**a**) and (**b**) are the input gene tree \(G\) and input species tree \(S\), respectively. (**c**) An augmented version of \(G\) containing a single hidden node as required by the reconciliations shown in Parts (**e,f**).