We establish a general bijective framework for encoding faces of some classical hyperplane arrangements. Precisely, we consider hyperplane arrangements in Rⁿ whose hyperplanes are all of the form \xᵢ-xⱼ=s\ for some i, j and s Z. Such an arrangement A is strongly transitive if it satisfies the following condition: if \xᵢ-xⱼ=s\ A and \xⱼ-xₖ=t\ A for some i, j, k n and s, t 0, then \xᵢ-xₖ=s+t\ A. For any strongly transitive arrangement A, we establish a bijection between the faces of A and some set of decorated plane trees.
Olivier Bernardi (Sat,) studied this question.