We review the notion of a [Formula: see text], an algebraic structure introduced recently by López, Préville-Ratelle and Ronco during their work on the splitting of associativity via [Formula: see text]-Dyck paths, and we also introduce Rota[Formula: see text]-algebras: both structures can be considered as generalizations of dendriform structures. We obtain examples of Dyck[Formula: see text]-algebras in terms of planar rooted binary trees equipped with a particular type of Rota–Baxter operator, and we present examples of Rota m -algebras using left averaging morphisms. As an application, we observe that the structures presented here allow us to introduce quite naturally a “non-associative version” of the Kadomtsev–Petviashvili hierarchy.