We consider the problem of reconfiguring non-crossing spanning trees on point sets. For a set $P$ of $n$ points in general position in the plane, the flip graph $F(P)$ has a vertex for each non-crossing spanning tree on $P$ and an edge between any two spanning trees that can be transformed into each other by the exchange of a single edge. This flip graph has been intensively studied, lately with an emphasis on determining its diameter diam$(F(P))$ for sets $P$ of $n$ points in convex position. The current best bounds are $\frac{14}{9}n-O(1) \leq$ diam$(F(P))<\frac{15}{9}n-3$ [Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber; SODA 2025]. The crucial tool for both the upper and lower bound are so-called *conflict graphs*, which the authors stated might be the key ingredient for determining the diameter (up to lower-order terms). In this paper, we pick up the concept of conflict graphs and show that this tool is even more versatile than previously hoped. As our first main result, we use conflict graphs to show that computing the flip distance between two non-crossing spanning trees is NP-hard, even for point sets in convex position. Interestingly, the result still holds for more constrained flip operations, concretely, compatible flips (where the removed and the added edge do not cross) and rotations (where the removed and the added edge share an endpoint). Extending the line of research from [BKUV SODA25], we present new insights on the diameter of the flip graph. Their lower bound is based on a constant-size pair of trees, one of which is *stacked*. We show that if one of the trees is stacked, then the lower bound is indeed optimal up to a constant term, that is, there exists a flip sequence of length at most $\frac{14}{9}(n-1)$ to any other tree. Lastly, we improve the lower bound on the diameter of the flip graph $F(P)$ for $n$ points in convex position to $\frac{11}{7}n-o(n)$.

翻译：我们考虑点集上非交叉生成树的重构问题。对于平面中处于一般位置的 $n$ 个点构成的集合 $P$，翻转图 $F(P)$ 的每个顶点对应 $P$ 上的一棵非交叉生成树，而当两棵生成树可通过交换单条边相互转化时，它们之间有一条边相连。该翻转图已被深入研究，近期的工作聚焦于确定当点集 $P$ 处于凸位置时其直径 diam$(F(P))$ 的值。当前最优的上下界为 $\frac{14}{9}n-O(1) \leq$ diam$(F(P))<\frac{15}{9}n-3$ [Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber; SODA 2025]。上下界证明的关键工具是所谓的*冲突图*，原文作者指出它可能是确定直径（忽略低阶项）的核心要素。本文沿用冲突图的概念，并证明该工具比此前预期的更具普适性。作为首个主要结果，我们利用冲突图证明：计算两棵非交叉生成树之间的翻转距离是NP-困难的，即便对于凸位置点集也是如此。有趣的是，该结论在更受限的翻转操作下依然成立，具体包括相容翻转（被移除与被添加的边不相交）与旋转操作（被移除与被添加的边共享一个端点）。延续 [BKUV SODA25] 的研究路线，我们提出了关于翻转图直径的新见解。他们的下界基于一对固定大小的树，其中一棵是*堆叠树*。我们证明：若其中一棵为堆叠树，则该下界在常数项意义下是最优的，即存在一条长度至多为 $\frac{14}{9}(n-1)$ 的翻转序列能到达任意其他生成树。最后，我们将凸位置 $n$ 点集翻转图 $F(P)$ 的直径下界改进至 $\frac{11}{7}n-o(n)$。