# Combining Orthology and Xenology Data in a Common Phylogenetic Tree

A rooted tree T with vertex labels t(v) and set-valued edge labels λ(e) defines maps δ and ε on the pairs of leaves of T by setting δ(x,y)=q if the last common ancestor lca(x,y) of x and y is labeled q, and m∈ε(x,y) if m∈λ(e) for at least one edge e along the path from lca(x,y) to y. We show that a pair of maps (δ,ε) derives from a tree (T,t,λ) if and only if there exists a common refinement of the (unique) least-resolved vertex labeled tree (T_δ,t_δ) that explains δ and the (unique) least resolved edge labeled tree (T_ε,λ_ε) that explains ε (provided both trees exist). This result remains true if certain combinations of labels at incident vertices and edges are forbidden.

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