We discuss adaptive, hierarchical, spatial subdivisions of planar polygonal maps. These maps are extensively used in cartography, GIS, computer graphics and computer vision. Our results are applicable to many subdivision schemes, including quad trees, k-d trees, and binary space partitioning (BSP). We focus on cell splitting rules and leaf organization methods, introducing and comparing three of them. The simple rule results in short hierarchy trees but leaf operations are costly. The convex rule enjoys efficient leaf operations at the expense of taller trees. The convex differences rule combines both short hierarchies and efficient operations to create the convex differences hierarchy (CDH). It utilizes the convex differences tree (CDT) representation of simple polygons. We also present vertex enlargement, a technique for reducing the map complexity which results in shorter trees and simplified splitting rules.