Advances and applications of the closest-tree algorithm and Hadamard conjugation in phylogenetic inference

In phylogenetic inference Hadamard methods and the closest-tree algorithm have been a promising alternative to likelihood-based methods. However, applications to actual biological problems have been limited so far. In the early nineties, Hendy and Penny (1993) developed the two-state closest-tree algorithm for estimating the optimal branch lengths of a phylogenetic tree, whose parameters correspond to the Cavenders molecular evolution model (CFN). Steel et al. (1992) then developed the four-state version of this method, whose parameters correspond to the Kimura 3STs molecular evolution model (K3ST). In both cases, formulas for solving the optimization problems were provided. Here, we do not only contribute with proofs for these formulas, but we also adapt this methodology to the orchid genus Lophiarella, whose phylogenetic relationships remain unclear. With this biological application, we show the efficacy of the closest-tree algorithm coupled with Hadamard conjugation, phylogenetic invariants and edge-parameter inequalities (in Fourier coordinates) in jointly inferring the tree topology and the molecular evolution model that best explains the data. Finally, we reconcile this phylogeny with biogeographical and morphological aspects within this genus.