An improved lower bound on the maximum number of non-crossing spanning trees

Abstract

We address the problem of counting geometric graphs on point sets. Using analytic combinatorics we show that the so-called double chain point configuration of N points has Ω* (12.31 N ) non-crossing spanning trees and Ω* (13.40 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 Toth in 2011. A new upper bound of O* (22.12 N ) for the number of non-crossing spanning trees of the double chain is also obtained.