On the number of spanning trees a planar graph can have

We prove that any planar graph on n vertices has less than O(5.2852ⁿ) spanning trees. Under the restriction that the planar graph is 3-connected and contains no triangle and no quadrilateral the number of its spanning trees is less than O(2.7156ⁿ). As a consequence of the latter the grid size needed to realize a 3d polytope with integer coordinates can be bounded by O(147.7ⁿ). Our observations imply improved upper bounds for related quantities: the number of cycle-free graphs in a planar graph is bounded by O(6.4884ⁿ), the number of plane spanning trees on a set of n points in the plane is bounded by O(158.6ⁿ), and the number of plane cycle-free graphs on a set of n points in the plane is bounded by O(194.7ⁿ).