We consider two fundamental computational problems that arise when comparing phylogenetic trees, rooted or unrooted, with non-identical leaf sets. The first problem arises when comparing two trees where the leaf set of one tree is a proper subset of the other. The second problem arises when the two trees to be compared have only partially overlapping leaf sets. The traditional approach to handling these problems is to first restrict the two trees to their common leaf set. An alternative approach that has shown promise is to first complete the trees by adding missing leaves, so that the resulting trees have identical leaf sets. This requires the computation of an optimal completion that minimizes the distance between the two resulting trees over all possible completions. We provide optimal linear-time algorithms for both completion problems under the widely-used Robinson-Foulds (RF) distance measure. Our algorithm for the first problem improves the time complexity of the current fastest algorithm from quadratic (in the size of the two trees) to linear. No algorithms have yet been proposed for the more general second problem where both trees have missing leaves. We advance the study of this general problem by proposing a biologically meaningful restricted version of the general problem and providing optimal linear-time algorithms for the restricted version. Our experimental results on biological data sets suggest that using completion-based RF distances can result in different evolutionary inferences compared to traditional RF distances.
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