This paper studies a general method for the numerical integration of ordinary differential equations. The method, defined in part 1, contains many known processes as special case, such as multistep methods, Runge-Kutta methods (multistage), Taylor, series (multiderivative) and their extensions (section 2). After a short section on trees and pairs of trees we derive formulas for the conditions to be satisfied by the free parameters in order to equalize the numerical approximation with the solution up to a certain order. Next we extend the reuslts of Kastlunger [6]. The proof given here is shorter than the original one. Finally we discuss formulas, with the help of which the conditions for the parameters can be reduced considerably and give numerical examples.