A subset of vertices is a maximum independent set if no two of the vertices are adjacent and the subset has maximum cardinality. A subset of vertices is called a maximum dissociation set if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. Zito proved that the maximum number of maximum independent sets of a tree of order $n$ is $2^{\frac{n-3}{2}}$ if $n$ is odd, and $2^{\frac{n-2}{2}}+1$ if $n$ is even and also characterized all extremal trees with the most maximum independent sets, which solved a question posed by Wilf. Inspired by the results of Zito, in this paper, by establishing four structure theorems and a result of $k$‐König–Egerváry graph, we show that the maximum number of maximum dissociation sets in a tree of order $n$ is

$$\{\begin{array}{cc}{3}^{\frac{n}{3}-1}+\frac{n}{3}+1\hfill & \text{if}n\equiv 0\left(\mathrm{mod}3\right),\hfill \\ 3^{\frac{n-1}{3}-1}+1\hfill & \text{if}n\equiv 1\left(\mathrm{mod}3\right),\hfill \\ 3^{\frac{n-2}{3}-1}\hfill & \text{if}n\equiv 2\left(\mathrm{mod}3\right),\hfill \end{array}$$

and also give complete structural descriptions of all extremal trees on which these maxima are achieved.

中文翻译：

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树中最大解离集的最大数量

如果没有两个顶点相邻并且该子集具有最大基数，则该子集的顶点是最大独立集。如果顶点的一个子集诱导一个顶点度最大为1的子图，并且该子集具有最大基数，则将其称为最大离解集。Zito证明了一个命令树的最大独立集的最大数目$ñ$ 是 $2^{\frac{ñ-3}{2}}$ 如果 $ñ$ 很奇怪 $2^{\frac{ñ-2}{2}}+\mathrm{1个}$ 如果 $ñ$也是均匀的，并且具有所有集合的最大极值树，这解决了威尔夫提出的问题。受Zito结果的启发，本文通过建立四个结构定理和$ķ$‐König–Egerváry图，我们显示了顺序树中最大解离集的最大数目 $ñ$ 是

$$\{\begin{array}{cc}{3}^{\frac{ñ}{3}-\mathrm{1个}}+\frac{ñ}{3}+\mathrm{1个}\hfill & \text{如果}ñ\equiv 0（模3），\hfill \\ 3^{\frac{ñ-\mathrm{1个}}{3}-\mathrm{1个}}+\mathrm{1个}\hfill & \text{如果}ñ\equiv \mathrm{1个}（模3），\hfill \\ 3^{\frac{ñ-2}{3}-\mathrm{1个}}\hfill & \text{如果}ñ\equiv 2（模3），\hfill \end{array}$$

并给出了达到这些最大值的所有极树的完整结构描述。