  The number of topologically different plane real algebraic curves of a given
degree $d$ has the form $\exp(C d^2 + o(d^2))$. We determine the best available
upper bound for the constant $C$. This bound follows from Arnold inequalities
on the number of empty ovals. To evaluate its rate we show its equivalence with
the rate of growth of the number of trees half of whose vertices are leaves and
evaluate the latter rate.
