The method of minimum evolution

The use of minimal spanning networks in phylogenetic studies seems to have been first mentioned by Edwards & Cavalli-Sforza (1963). They considered the construction of such networks in a general Euclidean space for their ‘method of minimum evolution’, which they introduced as an intuitive solution to the problem of reconstructing evolutionary trees from sample gene frequencies in present-day populations. The model underlying the suggestion is one of successive branching into two populations with random genetic drift then causing divergence between the two populations, increasing with the length of time since the existence of a common ancestor. It was suggested that the minimum-length network might be a reasonable approximation to the maximum-likelihood phylogenetic tree under this model. Of the phylogenetic studies that have been made using the method of minimal spanning networks only two fulfil the conditions of negligible population interaction and no hybridization for which the model waB suggested. Cavalli-Sforza & Edwards (1964) examine the case of 15 world populations, and Fitch & Neel ( 1969) consider the phylogenetic relationships between American Indian tribes. However, the method has also been used simply as a method of demonstrating relationships between populations in cases where there is a considerable degree of population interaction, and the model of divergence from a common ancestor caused by random genetic drift is not valid. The method of minimal spanning networks has proved successful in representing the relationships between tribal villages in that the results agree well with the known relationships on the basis of migration, geographical position, linguistics, history, etc. (Ward & Neel, 1970; Friedlaender et al. 1971). The results obtained using different sets of loci for the same populations also agree quite well with each other (Ward & Neel, 1970). Kidd & Sgaramella-Zonta ( 1971) review various network models including minimum evolution and compare the results obtained using the various methods on some actual data, but no detailed consideration of the methodology of minimal spanning networks has been previously published. The problem of connecting up a group of points in ;I plane appears first in the work of Steiner (see Courant & Robbins, 19G1), who Considered the problem of uniting three villages with a road net of minimum length, and proved that the minimal configuration is three branches mutually inclined a t 120’. Some details of a practical approach to the more general problem of connecting points in a multidimensional Euclidean space are given by Cavalli-Sforza & Edwards (1967). The problem has appeared in the literature recently from time to time (see, in particular, Gilbert & Pollack, 1968), mostly with reference to points in a plane rather than the multidimensional version, but no explicit method of solution is known in general. Following the terminology of Gilbert & Pollack (1968), we shall call a shortest interconnecting network spanning nodes A,, . , An a Steiner minimal tree. Junctions of branches of the network, other than those located at A, , . , A,, are known as internal nodes. A branch of the network