Transforming spanning trees: A lower bound

For a planar point set we consider the graph of crossing-free straight-line spanning trees where two spanning trees are adjacent in the graph if their union is crossing-free. An upper bound on the diameter of this graph implies an upper bound on the diameter of the flip graph of pseudo-triangulations of the underlying point set. We prove a lower bound of Ω ( log(n)/ log(log(n)) ) for the diameter of the graph of spanning trees on a planar set of n points. This nearly matches the known upper bound of O(log(n)). If we measure the diameter in terms of the number of convex layers k of the point set, our lower bound construction is tight, i.e., the diameter is in Ω(log(k)) which matches the known upper bound of O(log(k)). So far only constant lower bounds were known.