  We present a new method to count unrooted maps on the sphere up to
orientation-preserving homeomorphisms. The principle, called
tree-decomposition, is to deform a map into an arborescent structure whose
nodes are occupied by constrained maps. Tree-decomposition turns out to be very
efficient and flexible for the enumeration of constrained families of maps. In
this article, the method is applied to count unrooted 2-connected maps and,
more importantly, to count unrooted 3-connected maps, which correspond to the
combinatorial types of oriented convex polyhedra. Our method improves
significantly on the previously best-known complexity to enumerate unrooted
3-connected maps.
