A non-crossing spanning tree of a set of points in the plane is a spanning tree whose edges pairwise do not cross. Avis and Fukuda in 1996 proved that there always exists a flip sequence of length at most $2n-4$ between any pair of non-crossing spanning trees (where $n$ denotes the number of points). Hernando et al. proved that the length of a minimal flip sequence can be of length at least $\frac 32 n$. Two recent results of Aichholzer et al. and Bousquet et al. improved the Avis and Fukuda upper bound by proving that there always exists a flip sequence of length respectively at most $2n - \log n$ and $2n - \sqrt{n}$. We improve the upper bound by a linear factor for the first time in 25 years by proving that there always exists a flip sequence between any pair of non-crossing spanning trees $T_1,T_2$ of length at most $c n$ where $c \approx 1.95$. Our result is actually stronger since we prove that, for any two trees $T_1,T_2$, there exists a flip sequence from $T_1$ to $T_2$ of length at most $c |T_1 \setminus T_2|$. We also improve the best lower bound in terms of the symmetric difference by proving that there exists a pair of trees $T_1,T_2$ such that a minimal flip sequence has length $\frac 53 |T_1 \setminus T_2|$, improving the lower bound of Hernando et al. by considering the symmetric difference instead of the number of vertices. We generalize this lower bound construction to non-crossing flips (where we close the gap between upper and lower bounds) and rotations.

翻译：平面上点集的无交叉生成树是指边两两不相交的生成树。Avis与Fukuda于1996年证明，任意两棵无交叉生成树之间总存在长度不超过$2n-4$的翻转序列（其中$n$表示点数）。Hernando等人证明最小翻转序列的长度至少为$\frac 32 n$。近期Aichholzer等人与Bousquet等人的两项研究改进了Avis-Fukuda上界，分别证明了总存在长度不超过$2n - \log n$和$2n - \sqrt{n}$的翻转序列。我们在25年来首次将上界改进为线性因子，证明任意两棵无交叉生成树$T_1,T_2$之间总存在长度不超过$c n$的翻转序列，其中$c \approx 1.95$。我们的结果实际上更强：对于任意两棵树$T_1,T_2$，存在从$T_1$到$T_2$且长度不超过$c |T_1 \setminus T_2|$的翻转序列。同时，我们改进了关于对称差的下界，证明存在一对树$T_1,T_2$使得最小翻转序列长度为$\frac 53 |T_1 \setminus T_2|$，通过用对称差替代顶点数改进了Hernando等人的下界。我们将此下界构造推广至无交叉翻转（填补了上下界间隙）和旋转操作。