An iterative approach from the standpoint of the additive hypothesis to the dendrogram problem posed by molecular data sets

Abstract

The problem of constructing a dendrogram depicting phylogenetic relationships for a collection of contemporary species is considered. An approach was developed based on the additive hypothesis in which each "length" between two species can be described by the shortest sum of lengths for the individual links on the dendrogram topology which connect the two species. The additive hypothesis holds equally well if the dendro gram is replaced by its corresponding (rootless) network. Network topologies are defined set theoretically in terms of the initial, contemporary species, and a coefficient is defined for each point of any conceivable network. It is proved mathematically that each point of an additive network gives a coefficient value of zero, whereas each point not belonging to an additive network gives a coefficient value greater than zero. This suggests an iterative procedure in which "false" network points are replaced by "true" ones, or more generally in which "very false" network points are replaced by "nearly true" ones. The first procedure follows from the mathematical proof and the second is confirmed by simulation. Since most real data sets are not additive in the strict sense, a real data example was presented in which the iterative procedure produced a plausible network topology.