  We study the maximal number of triangulations that a planar set of $n$ points
can have, and show that it is at most $30^n$. This new bound is achieved by a
careful optimization of the charging scheme of Sharir and Welzl (2006), which
has led to the previous best upper bound of $43^n$ for the problem.
  Moreover, this new bound is useful for bounding the number of other types of
planar (i.e., crossing-free) straight-line graphs on a given point set.
Specifically, we derive new upper bounds for the number of planar graphs
($o(239.4^n)$), spanning cycles ($O(70.21^n)$), and spanning trees ($160^n$).
