We consider spanning trees of n points in convex position whose edges are
pairwise non-crossing. Applying a flip to such a tree consists in adding an
edge and removing another so that the result is still a non-crossing spanning
tree. Given two trees, we investigate the minimum number of flips required to
transform one into the other. The naive 2n−Ω(1) upper bound stood for 25
years until a recent breakthrough from Aichholzer et al. yielding a
2n−Ω(logn) bound. We improve their result with a 2n−Ω(n)
upper bound, and we strengthen and shorten the proofs of several of their
results