Hereditarily Optimal Realizations: Why are they Relevant in Phylogenetic Analysis, and how does one Compute them

One of the main problems in phylogenetic analysis (where one is concerned with elucidating evolutionary patterns between present day species) is to find good approximations of genetic distances by weighted trees. As an aid to solving this problem, it might seem tempting to consider an optimal realization of the metric defined by the given distances — the guiding principle being that, in case the metric is tree-like, the optimal realization obtained will necessarily be that unique weighted tree that realizes this metric. Although optimal realizations of arbitrary distances are not generally trees, but rather weighted graphs, one could still hope to obtain an informative representation of the given metric, maybe even more informative than the best approximating tree. However, optimal realizations are not only difficult to compute, they may also be non-unique. In this note we focus on one possible way out of this dilemma: hereditarily optimal realizations. These are essentially unique, and can also be described in an explicit way. We define hereditarily optimal realizations, discuss some of their properties, and we indicate in particular why, due to recent results on the so-called T-construction of a metric space, it is a straight forward task to compute these realizations for a large class of phylogentically relevant metrics. The author thanks the New Zealand Marsden Fund for its support. The author thanks the Swedish Natural Science Research Council (NFR) for its support (grant# M12342-300).