Initial pre-algebras as a generalization of dendriform algebras

We continue the study of \emph{initial dialgebras} defined in~\cite{DMS2026}. For a binary operad $\text{Var}$ we define the class of initial pre-$\text{Var}$-algebras and the corresponding operad $\text{pre}{\text{Var}}^{\text{I}}$ in such a way that $$(\text{di}{\text{Var}}^{\text{I}}{)}^{!}=(\text{pre}({\text{Var}}^{!}){)}^{\text{I}}$$ in the case when $\text{Var}$ is quadratic. We propose an intuitive algorithm for finding the defining relations of the operad $\text{pre}{\text{Var}}^{\text{I}}$ in the case when $\text{Var}$ is a binary quadratic operad. We also study free initial pre-algebras in the associative and commutative settings. For the nonsymmetric operad $\text{pre}{\text{As}}^{\text{I}}$, we construct a Grobner–Shirshov basis in the free magma operad, describe a linear basis in terms of admissible decorated planar binary trees, and establish a bijection between these trees and certain combinatorial objects.