A subset of vertices is a {it maximum independent set} if no two of the vertices are adjacent and the subset has maximum cardinality. A subset of vertices is called a {it maximum dissociation set} if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. Zito [J. Graph Theory {bf 15} (1991) 207--221] 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 $k$-K{o}nig-Egerv{a}ry graph, we show that the maximum number of maximum dissociation sets in a tree of order $n$ is begin{center} $left{ begin{array}{ll} 3^{frac{n}{3}-1}+frac{n}{3}+1, & hbox{if $nequiv0pmod{3}$;} 3^{frac{n-1}{3}-1}+1, & hbox{if $nequiv1pmod{3}$;} 3^{frac{n-2}{3}-1}, & hbox{if $nequiv2pmod{3}$,} end{array} right.$ end{center} and also give complete structural descriptions of all extremal trees on which these maxima are achieved.
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