Understanding when an abstract complex curve C of genus g comes equipped with a nondegenerate degree-d morphism to P^r is a fundamental question for curve theory. When C is general in moduli, and answer is provided by the celebrated Brill--Noether theorem, which establishes that the space of morphisms on C behaves as one would expect on the basis of linear algebra. On the other hand, the ``dual graph" construction provides a bridge between the moduli space of curves and the moduli space of (stable) metric graphs. This leads to the question of which stable graphs are general in a Brill--Noether sense. In work in progress with Melo, Neves, and Viviani, we study a seemingly-remarkable family of examples of graphs that decompose as triples of trees rooted on a set of d common vertices. Here we will describe some of the combinatorics involved, and how these graphs relate to the configuration spaces n points on P^1. |