I show new bijective proofs of previously known formulas for the number of regions of some deformations of the braid arrangement, by means of a bijection between the no-broken-circuit sets of the corresponding integral gain graphs and some kinds of labelled binary trees. This leads to new bijective proofs for the Shi, Catalan, and similar hyperplane arrangements. It may lead to interesting new properties of the Linial arrangement.
This is joint work with Sylvie Corteel and Véronique Ventos.