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Gene Tree Diameter for Deep Coalescence

The deep coalescence cost accounts for discord caused by deep coalescence between a gene tree and a species tree. It is a major concern that the diameter of a gene tree (the tree's maximum deep coalescence cost across all species trees) depends on its topology, which can largely obfuscate phylogenetic studies. While this bias can be compensated by normalizing the deep coalescence cost using diameters, obtaining them efficiently has been posed as an open problem by Than and Rosenberg [33]. Here, we resolve this problem by describing a linear time algorithm to compute the diameter of a gene tree. In addition, we provide a complete classification of the species trees yielding this diameter to guide phylogenetic analyses.