Gröbner–Shirshov bases for the non-symmetric operads of dendriform algebras and quadri-algebras

In this paper we use the operadic framework to find Gröbner–Shirshov bases for the free quadri-algebra. We perform computations using the representation of the non-symmetric operad by planar rooted trees in a very intuitive way. Gröbner–Shirshov bases for the free dendriform algebra are also found with this technique, simplifying the work by Chen and Wang (2010).