  For a simple drawing $D$ of the complete graph $K_n$, two (plane) subdrawings
are compatible if their union is plane. Let $\mathcal{T}_D$ be the set of all
plane spanning trees on $D$ and $\mathcal{F}(\mathcal{T}_D)$ be the
compatibility graph that has a vertex for each element in $\mathcal{T}_D$ and
two vertices are adjacent if and only if the corresponding trees are
compatible. We show, on the one hand, that $\mathcal{F}(\mathcal{T}_D)$ is
connected if $D$ is a cylindrical, monotone, or strongly c-monotone drawing. On
the other hand, we show that the subgraph of $\mathcal{F}(\mathcal{T}_D)$
induced by stars, double stars, and twin stars is also connected. In all cases
the diameter of the corresponding compatibility graph is at most linear in $n$.
