Computing Covers of Plane Forests

Let φ be a function that maps any non-empty subset A of R to a non-empty subset φ(A) of R. 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), where A and B are constructed by either 2a or 2b, and A ∩B 6= ∅. We present two properties for the function φ that make the φ-cover well-defined. Examples for such functions φ are the convex hull and the axis-aligned bounding box. For both of these functions φ, we show that the φ-cover can be computed in O(n log n) time, where n is the total number of vertices of the trees in T .