There are important reasons to be sceptical of the accuracy and usefulness of the family-tree model in historical linguistics. That model assumes that every linguistic innovation applies to a language considered as an undifferentiated whole, a point with no “width”. But this assumption makes it impossible to use a tree to model the partial diffusion of an innovation within a language community (“internal diffusion”), or the diffusion of an innovation across language communities (“external diffusion”). These limitations have long been noticed by historical linguists (Schmidt 1872, Schuchardt 1900); but they become glaringly obvious in the cases discussed by Ross (1988) and François (2014) under the heading of “linkages” – i.e., language families that arise through the diversification, in situ, of a dialect network. The articles in this special issue all contribute towards addressing this problem, from a range of perspectives.