Packing plane spanning trees into a point set

Let $P$ be a set of $n$ points in the plane in general position. We show that at least $\lfloor n/3\rfloor$ plane spanning trees can be packed into the complete geometric graph on $P$. This improves the previous best known lower bound $\Omega\left(\sqrt{n}\right)$. Towards our proof of this lower bound we show that the center of a set of points, in the $d$-dimensional space in general position, is of dimension either $0$ or $d$.