Initial value problems of autonomous systems of ordinary differential equations are considered, where the right-hand side is split into a linear and a nonlinear part, respectively. The coefficients of a common Runge-Kutta method yield a class of exponential integrators applied to the system. Following \textit{E. Hairer, S. P. Nørsett}, and \textit{G. Wanner} [Solving ordinary differential equations. I, 2nd rev. ed., Springer Series in Computational Mathematics. 8. Berlin: Springer-Verlag. (1993; Zbl 0789.65048)], the theory of B-series describes the order conditions for consistency of the method. Bicoloured rooted trees arise due to the splitting in linear and nonlinear part. A convenient approach consists in choosing finite dimensional function spaces, where the coefficient functions of the exponential integrator will be situated. Furthermore, a class of methods is generated by approximating the nonlinear part uniformly by a polynomial. \textit{M. Zennaro} [Math. Comput. 46, 119--133 (1986; Zbl 0608.65043)] introduced a natural continuous extension in case of arbitrary Runge-Kutta schemes. The authors apply this idea to define a natural continuous extension, which can be used to construct exponential integrators. It follows that the degree of the extension determines the order of the corresponding exponential integrator. Moreover, a natural continuous extension exists for each underlying Runge-Kutta method. A crucial result of the paper is that the dimensions of specific function spaces, which are used to determine the coefficient functions, feature lower bounds by terms depending on the order of a considered exponential integrator. The authors apply a technique based on bicoloured trees to prove the formulas. The bounds are not shown to be sharp, but provide a deeper insight in the structure of the methods. Examples of according exponential integrators of order 4 and 5 are presented. Numerical simulations are not within the scope of the paper. Alternatively, the paper provides an analysis of consistency conditions corresponding to a class of exponential integrators.