  Bernardi has given a general formula for the number of regions of a
deformation of the braid arrangement as a signed sum over boxed trees. We prove
that 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 for
computing this value. For Ish-type arrangements, we further construct a
sign-reversing involution which reduces Bernardi's signed sum to the
enumeration of a set of (rooted labeled plane) trees. We conclude by explicitly
enumerating the trees corresponding to the regions of Ish-type arrangements
which are nested, recovering their known counting formula.
