The string repetitiveness measures $χ$ (the size of a smallest suffixient set of a string) and $r$ (the number of runs in the Burrows--Wheeler Transform) are related. Recently, we have shown that the bound $χ\leq 2r$, proved by Navarro et al., is asymptotically tight as the size $σ$ of the alphabet increases, but achieving near-tight ratios for fixed $σ> 2$ remained open. We introduce a \emph{2-branching property}: a cyclic string is 2-branching at order~$k$ if every $(k{-}1)$-length substring admits exactly two $k$-length extensions. We show that 2-branching strings of order~$k$ yield closed-form ratios $χ/r = (2σ^{k-1}+1)/(σ^{k-1}+4)$. For order~$3$, we give an explicit construction for every $σ\geq 2$, narrowing the gap to~$2$ from $O(1/σ)$ to $O(1/σ^2)$. For $σ\in \{3,4\}$, we additionally present order-$5$ instances with ratios exceeding~$1.91$.

翻译：字符串可重复性度量$χ$（字符串的最小后缀集合的大小）与$r$（Burrows--Wheeler变换中的游程数）是相关的。最近，我们证明了Navarro等人提出的界限$χ\leq 2r$在字母表大小$σ$增大时是渐近紧的，但对于固定的$σ> 2$，实现接近紧的比率仍然是一个开放问题。我们引入了\emph{2-分支性质}：一个循环字符串在阶$k$下是2-分支的，如果每个长度为$(k{-}1)$的子串恰好有两个长度为$k$的扩展。我们证明了阶$k$的2-分支字符串可以产生闭式比率$χ/r = (2σ^{k-1}+1)/(σ^{k-1}+4)$。对于阶$3$，我们为每个$σ\geq 2$给出了一个显式构造，将比率与$2$的差距从$O(1/σ)$缩小到$O(1/σ^2)$。对于$σ\in \{3,4\}$，我们还额外给出了阶$5$的实例，其比率超过$1.91$。