Abstract. The neighbor-joining algorithm is a popular phylogenetics method for constructing trees from dissimilarity maps. The neighbor-net algorithm is an extension of the neighbor-joining algorithm and is used for constructing split networks. We begin by describing the output of neighbor-net in terms of the tessellation of M n 0 by associahedra. This highlights the fact that neighbor-net outputs a tree in addition to a circular ordering and we explain when the neighbor-net tree is the neighbor-joining tree. A key observation is that the tree constructed in existing implementations of neighbor-net is not a neighbor-joining tree. Next, we show that neighbor-net is a greedy algorithm for finding circular split systems of minimal balanced length. This leads to an interpretation of neighbor-net as a greedy algorithm for the traveling salesman problem. The algorithm is optimal for Kalmanson matrices, from which it follows that neighbor-net is consistent and has optimal radius 1 2. We also provide a statistical interpretation for the balanced length for a circular split system as the length based on weighted least squares estimates of the splits. We conclude with applications of these results and demonstrate the implications of our theorems for a recently published comparison of Papuan and Austronesian languages. 1.

Instead of considering scales to be linearly ordered structures, it is proposed that scales are better conceived of as metrics (dissimilarity matrices). Further, to be considered a scale of typological interest, there should be a significant correlation between a meaning-scale and a form-scale. This conceptualisation allows for a fruitful generalization of the concept "scale". As a hands-on example of the proposals put forward in this paper, the "scale of likelihood of spontaneous occurrence " (Haspelmath 1993) is reanalyzed. This scale describes the prototypical agentivity of the subject of a predicate. 1. Scales as restrictions on form-function mapping Scales 1 of linguistic structure are one of the more promising avenues of research into the unification of the worldwide linguistic diversity. Although our growing understanding of the diversity of the world’s languages seems to put more and more doubt on many grandiose attempts on universally valid generalizations, the significance of scales for human languages (like the well-known animacy scale) still appears to stand strong. So, what actually is a scale? A scale seems to be mostly thought of as an asymmetrical one-dimensional arrangement (a “total order ” in mathematical parlance) on certain cross-linguistic categories/functions. Put differently, a scale is a linear ordering of functions with a “high end ” and a “low end”. To be a considered an interesting scale, the formal encoding of these functions in actual languages should be related to this linear ordering. In this paper, I will argue that this concept of a scale can be fruitfully generalized. In a very general sense, all linguistic structure consists of forms expressing particular functions. If we find restrictions—across languages— on the kind of forms that are used to express certain functions, then this 1 The term “scale ” is used here synonymously to what is also known as an “implicational hierarchy”, “markedness hierarchy ” or simply “hierarchy ” in linguistics.