  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 $2n-\Omega(1)$ upper bound stood for 25
years until a recent breakthrough from Aichholzer et al. yielding a
$2n-\Omega(\log n)$ bound. We improve their result with a $2n-\Omega(\sqrt{n})$
upper bound, and we strengthen and shorten the proofs of several of their
results.
