It follows from a classical result of Jordan that every tree with maximum degree at most $r$ containing a vertex set labeled by $[n]$, has a single-edge cut which separates two subsets $A,B \subset [n]$ for which $\min\{|A|,|B|\} \ge (n-1)/r$. Motivated by the tree dissimilarity problem in phylogenetics, we consider the case of separating vertex sets of {\em several} trees: Given $k$ trees with maximum degree at most $r$, containing a common vertex set labeled by $[n]$, we ask for a single-edge cut in each tree which maximizes $min\{|A|,|B|\}$ where $A,B \subset [n]$ are separated by the corresponding cut at each tree. Denoting this maximum by $f(r,k,n)$ and considering the limit $f(r,k) = \lim_{n \rightarrow \infty} f(r,k,n)/n$ (which is shown to always exist) we determine that $f(r,2)=\frac{1}{2r}$ and determine that $f(3,3)=\frac{2}{27}$, which is already quite intricate. The case $r=3$ is especially interesting in phylogenetics and our result implies that any two (three) binary phylogenetic trees over $n$ taxa have a split at each tree which separates two taxa sets of order at least $n/6$ (resp. $2n/27$), and these bounds are asymptotically tight.

翻译：根据Jordan的经典结果，每个最大度不超过$r$且包含由$[n]$标记的顶点集的树，都存在一条单边割，该割分离出两个子集$A,B \subset [n]$，满足$\min\{|A|,|B|\} \ge (n-1)/r$。受系统发育学中树相异性问题的启发，我们考虑分离多个树的顶点集的情形：给定$k$棵最大度不超过$r$且包含由$[n]$标记的公共顶点集的树，我们寻求每棵树中的一条单边割，使得在所有树中对应的割所分离出的$A,B \subset [n]$满足$\min\{|A|,|B|\}$最大化。记该最大值为$f(r,k,n)$，并考虑极限$f(r,k) = \lim_{n \rightarrow \infty} f(r,k,n)/n$（我们证明该极限始终存在），我们确定了$f(r,2)=\frac{1}{2r}$，并确定了$f(3,3)=\frac{2}{27}$，后者已相当复杂。$r=3$的情形在系统发育学中尤为有趣，我们的结果意味着任意两棵（三棵）基于$n$个分类群的二叉系统发育树，在每棵树中都存在一个分割，该分割分离出阶数至少为$n/6$（相应地，$2n/27$）的两个分类群集合，并且这些界是渐近紧的。