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Bernardi has given a general formula for the number of regions of adeformation of the braid arrangement as a signed sum over boxed trees. We provethat each set of boxed trees which share an underlying (rooted labeled plane)tree contributes 0, +1, or -1 to this sum, and we give an algorithm forcomputing this value. For Ish-type arrangements, we further construct asign-reversing involution which reduces Bernardi's signed sum to theenumeration of a set of (rooted labeled plane) trees. We conclude by explicitlyenumerating the trees corresponding to the regions of Ish-type arrangementswhich are nested, recovering their known counting formula.

The Bernardi formula for non-transitive deformations of the braid arrangement