Compatible Spanning Trees in Simple Drawings of $$K_n$$

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 set all plane spanning trees on and {F}(\mathcal {T}_D)$$ compatibility that has vertex for each element in vertices adjacent only corresponding compatible. We show, one hand, connected cylindrical, monotone, or strongly c-monotone drawing. On other we show subgraph induced by stars, double twin stars also connected. In cases diameter at most linear n.