# Improved Duplication-Transfer-Loss Reconciliation with
## 2 Definitions and Preliminaries
### DTLx Reconciliation  
A DTLx reconciliation of \(G\) and \(S\) shows how a suitably augmented version \(G^{\prime}\) of \(G\) evolved inside the augmented species tree \(S^{\prime}\) through speciation, gene duplication, transfer, \(\mathbb{TL}\), \(\mathbb{TX}\), and loss events. A formal definition of DTLx reconciliation appears below and it specifies what constitutes a biologically valid DTLx reconciliation of \(G\) and \(S\). **Definition 5** (DTLx Reconciliation). : _A DTLx reconciliation for \(G\) and \(S\) is an eleven-tuple \(\langle\mathcal{L},G^{\prime},S^{\prime},\mathcal{M},\Sigma,\Delta,\Theta, \Theta_{\mathrm{L}},\Theta_{\mathrm{X}},\Xi,\tau\rangle\), where \(\mathcal{L}:L\!e(G)\to L\!e(S)\) represents the leaf mapping from \(G\) to \(S\), \(G^{\prime}\) denotes the augmented gene tree, \(S^{\prime}\) denotes the augmented species tree, \(\mathcal{M}\): \(V(G^{\prime})\to V(S^{\prime})\) maps each node of \(G^{\prime}\) to a node of \(S^{\prime}\), the sets \(\Sigma\), \(\Delta\), \(\Theta\), \(\Theta_{\mathrm{L}}\), and \(\Theta_{\mathrm{X}}\) partition \(I(G^{\prime})\) into speciation, duplication, \(\mathbb{T}\), \(\mathbb{TL}\), and \(\mathbb{TX}\) events, respectively; \(\Xi\) is the subset of \(E(G^{\prime})\) that represents transfer edges, and \(\tau:\Theta\cup\Theta_{\mathrm{L}}\cup\Theta_{\mathrm{X}}\to V(S^{\prime})\) specifies the recipient species for each transfer event, subject to the following constraints:_  
**Augmented gene tree constraint**__  
1. \(G\) _can be obtained from_ \(G^{\prime}\) _by suppressing each node of_ \(G^{\prime}\) _with exactly one child._  
**Mapping constraints**__  
2. _If_ \(g\in L\!e(G^{\prime})\)_, then_ \(\mathcal{M}(g)=\mathcal{L}(g)\)  
Figure 3: Representation of unsampled lineages on the species tree. (**a**) Illustration of the “coral” of life based on drawings by [PERSON], where black lines represent lineages leading to extant species, and grey lines represent potential unsampled species; adapted from [28]. (**b**) Input species tree before augmentation. (**c**) Augmented species tree. Extra nodes are in green and extra leaves are in white. Leaves labeled A-D represent extant species. Unsampled lineages are represented by the edges connecting extra nodes and extra leaves. _._ 3. _If_ \(g\in I(G)\cap I(G^{\prime})\) _(i.e.,_ \(g\) _is not a hidden node of_ \(G^{\prime}\)_) and_ \(g^{\prime}\) _and_ \(g^{\prime\prime}\) _denote the children of_ \(g\) _in_ \(G^{\prime}\)_, then,_ 1. \(\mathcal{M}(g)\
ot<_{S^{\prime}}\mathcal{M}(g^{\prime})\)_, and_ \(\mathcal{M}(g)\
ot<_{S^{\prime}}\mathcal{M}(g^{\prime\prime})\)_,_ 2. _At least one of_ \(\mathcal{M}(g^{\prime})\) _and_ \(\mathcal{M}(g^{\prime\prime})\) _is a descendant of_ \(\mathcal{M}(g)\)_._ 4. _If_ \(g\in I(G^{\prime})\setminus I(G)\) _(i.e.,_ \(g\) _is a hidden node of_ \(G^{\prime}\)_) and_ \(g^{\prime}\) _denotes it's unique child, then_ \(\mathcal{M}(g)\) _and_ \(\mathcal{M}(g^{\prime})\) _are incomparable._  
**Event constraints**  
1. _Given any edge_ \((g,g^{\prime})\in E(G^{\prime})\)_,_ \((g,g^{\prime})\in\Xi\) _if and only if_ \(\mathcal{M}(g)\) _and_ \(\mathcal{M}(g^{\prime})\) _are incomparable._
2. _If_ \(g\in I(G)\cap I(G^{\prime})\) _and_ \(g^{\prime}\) _and_ \(g^{\prime\prime}\) _denote the children of_ \(g\) _in_ \(G^{\prime}\)_, then,_ 1. \(g\in\Sigma\) _only if_ \(\mathcal{M}(g)=\text{lca}(\mathcal{M}(g^{\prime}),\mathcal{M}(g^{\prime\prime}))\) _and_ \(\mathcal{M}(g^{\prime})\) _and_ \(\mathcal{M}(g^{\prime\prime})\) _are incomparable,_ 2. \(g\in\Delta\) _only if_ \(\mathcal{M}(g)\geq_{S^{\prime}}\text{lca}(\mathcal{M}(g^{\prime}),\mathcal{M}(g ^{\prime\prime}))\)_,_ 3. \(g\in\Theta\) _if and only if either_ \((g,g^{\prime})\in\Xi\) _or_ \((g,g^{\prime\prime})\in\Xi\)_,_ 4. _If_ \(g\in\Theta\) _and_ \((g,g^{\prime})\in\Xi\)_, then_ \(\mathcal{M}(g)\) _and_ \(\tau(g)\) _must be incomparable and_ \(\mathcal{M}(g^{\prime})\) _must be a descendant of_ \(\tau(g)\)_, i.e.,_ \(\mathcal{M}(g^{\prime})\leq_{S^{\prime}}\tau(g)\)_._
3.