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 - Ω(log n) bound. We improve their result with a 2n - Ω(√n) upper bound, and we strengthen and shorten the proofs of several of their results.