  This paper is a contribution to the problem of counting geometric graphs on
point sets. More concretely, we look at the maximum numbers of non-crossing
spanning trees and forests. We show that the so-called double chain point
configuration of N points has Omega(12.52^N) non-crossing spanning trees and
Omega(13.61^N) non-crossing forests. This improves the previous lower bounds on
the maximum number of non-crossing spanning trees and of non-crossing forests
among all sets of N points in general position given by Dumitrescu, Schulz,
Sheffer and T\'oth. Our analysis relies on the tools of analytic combinatorics,
which enable us to count certain families of forests on points in convex
position, and to estimate their average number of components. A new upper bound
of O(22.12^N) for the number of non-crossing spanning trees of the double chain
is also obtained.
