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.

翻译：我们考虑凸位置上 $n$ 个点的生成树，其边两两非交叉。对此类树进行翻转操作包括添加一条边并移除另一条边，使得结果仍为非交叉生成树。给定两棵树，我们研究将一棵树转化为另一棵树所需的最小翻转次数。原始的 $2n-\Omega(1)$ 上界持续了25年，直到近期Aichholzer等人取得突破，得到了 $2n-\Omega(\log n)$ 的上界。我们将其结果改进为 $2n-\Omega(\sqrt{n})$ 的上界，并加强和简化了其若干结果的证明。