Background: Phylogenetic tree comparison metrics are an important tool in the study of evolution, and hence the definition of such metrics is an interesting problem in phylogenetics. In a paper in Taxon fifty years ago, Sokal and Rohlf proposed to measure quantitatively the difference between a pair of phylogenetic trees by first encoding them by means of their half-matrices of cophenetic values, and then comparing these matrices. This idea has been used several times since then to define dissimilarity measures between phylogenetic trees but, to our knowledge, no proper metric on weighted phylogenetic trees with nested taxa based on this idea has been formally defined and studied yet. Actually, the cophenetic values of pairs of different taxa alone are not enough to single out phylogenetic trees with weighted arcs or nested taxa.Results: For every (rooted) phylogenetic treeT, let itscophenetic vectorφ(T) consist of all pairs of cophenetic values between pairs of taxa inTand all depths of taxa inT. It turns out that these cophenetic vectors single out weighted phylogenetic trees with nested taxa. We then define a family of cophenetic metricsd_φ,pby comparing these cophenetic vectors by means ofL^pnorms, and we study, either analytically or numerically, some of their basic properties: neighbors, diameter, distribution, and their rank correlation with each other and with other metrics.Conclusions: The cophenetic metrics can be safely used on weighted phylogenetic trees with nested taxa and no restriction on degrees, and they can be computed inO(n²) time, wherenstands for the number of taxa. The metricsd_φ,1andd_φ,2have positive skewed distributions, and they show a low rank correlation with the Robinson-Foulds metric and the nodal metrics, and a very high correlation with each other and with the splitted nodal metrics. The diameter ofd_φ,p, forp⩾1, is inO(n^(p+2)/p), and thus for lowpthey are more discriminative, having a wider range of values.

In this paper, the fuzzy morphological gradients from the fuzzy mathematical morphologies based on t-norms and conjunctive uninorms are deeply analyzed in order to establish which pair of conjunction and fuzzy implications are optimal, in accordance with their performance in edge detection applications. A novel three-step algorithm based on the fuzzy morphology is proposed. The comparison is performed by means of the so-called Pratt's figure of merit. In addition, a statistical analysis is carried out to study the relationship between the different configurations and to establish a classification of the conjunctions and implications considered. Both the objective measure and the statistical analysis conclude that the pairs nilpotent minimum t-norm and the Kleene-Dienes implication, and the idempotent uninorm obtained with the classical negation as a generator and its residual implication, are the best configurations in this approach, because they also obtain competitive results with respect to other approaches.