Henneberg  and Laman  characterized graphs which have, after adding any edge, 2 edge-disjoint spanning trees. This was generalized to $k$ edge-disjoint spanning trees by Frank and Szegõ . They also gave a characterization of graphs which have $k$ edge-disjoint spanning trees after the deletion of any edge of them. Furthermore, it is known that $k$ edge-disjoint spanning trees can be found in polynomial time (see e.g. ).
It is known that a minimum diameter spanning tree can also be found in polynomial time . My guess is that we can also find $k$ minimum diameter spanning trees efficiently. But is a characterization of graphs that have $k$ edge-disjoint minimum diameter spanning trees known, even for $k=2$?
 Henneberg, Lebrecht. Die graphische Statik der starren Systeme. Vol. 31. BG Teubner, 1911.
 Laman, Gerard. "On graphs and rigidity of plane skeletal structures." Journal of Engineering mathematics 4.4 (1970): 331-340.