Computing Covers of Plane Forests

Let $ϕ$ be a function that maps any non-empty subset $A$ of $\mathbb{R}^2$ to a non-empty subset $ϕ(A)$ of $\mathbb{R}^2$. A $ϕ$-cover of a set $T=\{T_1, T_2, \dots, T_m\}$ of pairwise non-crossing trees in the plane is a set of pairwise disjoint connected regions such that each tree $T_i$ is contained in some region of the cover, and each region of the cover is either (1) $ϕ(T_i)$ for some $i$, or (2) $ϕ(A \cup B)$, where $A$ and $B$ are constructed by either (1) or (2), and $A \cap B \neq \emptyset$. 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 $O(n\log^2n)$ time, where $n$ is the total number of vertices of the trees in $T$.

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