Computing covers of plane forests

Let φ be a function that maps any non-empty subset A of R2 to a non-empty subset φ(A) of R2. A φ cover of a set T = {T1, T2, . . . , Tm} of pairwise non-crossing trees in the plane is a set of pairwise disjoint connected regions such that 1. each tree Ti is contained in some region of the cover, 2. each region of the cover is either (a) φ(Ti) for some i, or (b) φ(A ∪ B), wh