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On the number of spanning trees a planar graph can have

We prove that any planar graph on $n$ vertices has less than $O(5{.}2852^n)$ 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 $O(2{.}7156^n)$. As a consequence of the latter the grid size needed to realize a 3d polytope with integer coordinates can be bounded by $O(147.{7}^n)$. Our observations imply improved upper bounds for related quantities: the number of cycle-free graphs in a planar graph is bounded by $O(6.4884^n)$, the number of plane spanning trees on a set of $n$ points in the plane is bounded by $O(158.6^n)$, and the number of plane cycle-free graphs on a set of $n$ points in the plane is bounded by $O(194{.}7^n)$.