Given a set $P$ of points in the plane, the geometric tree graph of $P$ is defined as the graph $T(P)$ whose vertices are non-crossing rectilinear spanning trees of $P$, and where two trees $T_1$ and $T_2$ are adjacent if $T_2 = T_1 -e+f$ for some edges $e$ and $f$. In this paper we concentrate on the geometric tree graph of a set of $n$ points in convex position, denoted by $G_n$. We prove several results about $G_n$, among them the existence of Hamilton cycles and the fact that they have maximum connectivity.