In this talk, for a given variety Var, we present a universal algorithm for constructing a subvariety of Var-dialgebras from which one can recover an algebra belonging to Var. Such a subvariety is called the variety of initial Var-dialgebras. We construct a basis of the free initial Lie and associative dialgebras. Also, for a binary operad Var, we define the class of initial pre-Var-algebras and the corresponding operad $preVar^I$ in such a way that $(diVar^I)^!=(pre(Var^!))^I$ in the case when Var is quadratic. We propose an intuitive algorithm for finding the defining relations of the operad $preVar^I$ in the case when Var is a binary quadratic operad. We also study free initial pre-algebras in the associative and commutative settings. For the nonsymmetric operad $preAs^I$, we construct a Gröbner-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.