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The maximum number of maximum dissociation sets in trees

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 2n−32 if n is odd, and 2n−22+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 {3n3−1+n3+1ifn≡0(mod3),3n−13−1+1ifn≡1(mod3),3n−23−1ifn≡2(mod3), and also give complete structural descriptions of all extremal trees on which these maxima are achieved.