Contrary to widespread belief, there is no reason to think that language diversification typically follows a tree-like pattern, consisting of a nested series of neat splits. Except for the odd case of language isolation or swift migration and dispersal, the normal situation is for language change to involve multiple events of diffusion across mutually intelligible idiolects in a network, typically distributed into conflicting isoglosses. Insofar as these events of language-internal diffusion are later reflected in descendant languages, the sort of language family they define - a "linkage" (Ross 1988) - is one in which genealogical relations cannot be represented by a tree, but only by a diagram in which subgroups intersect. Non-cladistic models are needed to represent language genealogy, in ways that take into account the common case of linkages and intersecting subgroups. This paper will focus on an approach that combines the precision of the Comparative Method with the realism of the Wave Model. This method, labeled Historical Glottometry, identifies genealogical subgroups in a linkage situation, and assesses their relative strengths based on the distribution of innovations among modern languages. Provided it is applied with the rigour inherent to the Comparative Method, Historical Glottometry should help unravel the genealogical structures of the world's language families, by acknowledging the role played by linguistic convergence and diffusion in the historical processes of language diversification.