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In a graph $G$, a subset of vertices is a dissociation set if it induces asubgraph with vertex degree at most 1. A maximum dissociation set is adissociation set of maximum cardinality. The dissociation number of $G$,denoted by $\psi(G)$, is the cardinality of a maximum dissociation set of $G$.Extremal problems involving counting the number of a given type of substructurein a graph have been a hot topic of study in extremal graph theory throughoutthe last few decades. In this paper, we determine the maximum number of maximumdissociation sets in a tree with prescribed dissociation number and theextremal trees achieving this maximum value.

On the maximum number of maximum dissociation sets in trees with given dissociation number