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 one edge flip. The diameter ${\rm diam}(F(P))$ of this graph is subject of intensive study. For points in general position, it is between $3n/2-5$ and $2n-4$, with no improvement for 25 years. For points in convex position, it lies between $3n/2 - 5$ and $\approx1.95n$, where the lower bound was conjectured to be tight up to an additive constant and the upper bound is a recent breakthrough improvement over several bounds of the form $2n-o(n)$. In this work, we provide new upper and lower bounds on ${\rm diam}(F(P))$, mainly focusing on points in convex position. We show $14n/9 - O(1) \le {\rm diam}(F(P)) \le 5n/3 - 3$, by this disproving the conjectured upper bound of $3n/2$ for convex position, and relevantly improving both the long-standing lower bound for general position and the recent new upper bound for convex position. We complement these by showing that if one of $T,T'$ has at most two boundary edges, then ${\rm dist}(T,T') \le 2d/2 < 3n/2$, where $d = |T-T'|$ is the number of edges in one tree that are not in the other. To prove both the upper and the lower bound, we introduce a new powerful tool. Specifically, we convert the flip distance problem for given $T,T'$ to the problem of a largest acyclic subset in an associated conflict graph $H(T,T')$. In fact, this method is powerful enough to give an equivalent formulation of the diameter of $F(P)$ for points $P$ in convex position up to lower-order terms. As such, conflict graphs are likely the key to a complete resolution of this and possibly also other reconfiguration problems.

翻译：对于平面上处于一般位置的 $n$ 个点构成的集合 $P$，其翻转图 $F(P)$ 的每个顶点对应于 $P$ 上的一棵非交叉生成树，若两棵生成树可通过一次边翻转操作相互转换，则它们对应的顶点之间存在一条边。该图的直径 ${\rm diam}(F(P))$ 是深入研究的课题。对于一般位置的点集，其直径介于 $3n/2-5$ 与 $2n-4$ 之间，且这一结果在25年间未得到改进。对于凸位置的点集，其直径介于 $3n/2 - 5$ 与 $\approx1.95n$ 之间，其中下界被猜想在相差一个加性常数的意义下是紧的，而上界是近期取得的突破性进展，它改进了多个形如 $2n-o(n)$ 的界。在本工作中，我们给出了 ${\rm diam}(F(P))$ 的新上界和新下界，主要聚焦于凸位置的点集。我们证明了 $14n/9 - O(1) \le {\rm diam}(F(P)) \le 5n/3 - 3$，由此否定了关于凸位置点集直径上界为 $3n/2$ 的猜想，并显著改进了长期存在的一般位置点集的下界以及近期提出的凸位置点集的新上界。作为补充，我们证明了若 $T,T'$ 中至少有一棵树的边界边数不超过两条，则 ${\rm dist}(T,T') \le 2d/2 < 3n/2$，其中 $d = |T-T'|$ 是一棵树中存在而另一棵树中不存在的边的数量。为了证明上界和下界，我们引入了一个新的强大工具。具体而言，我们将给定 $T,T'$ 之间的翻转距离问题，转化为在关联的冲突图 $H(T,T')$ 中寻找最大无圈子集的问题。事实上，该方法足够强大，能够给出凸位置点集 $P$ 对应的 $F(P)$ 直径的等价表述（精确到低阶项）。因此，冲突图很可能成为完全解决此问题以及可能其他重配置问题的关键。