The order conditions for any order (stiff) \(p\) on the coefficients of exponential integrators for the numerical solution of some stiff semilinear initial value problems with the stiffness included in the linear part are obtained. The problems under consideration can be written in the form: \( u' = F(u) \equiv A u + g(u)\), \( u(0)= u_0 \in X\), where \( u: [0, T] \to X\) is a sufficiently smooth function with derivatives in the Banach space \( ( X , \| \cdot \| )\), \( g : X \to X\) is also sufficiently differentiable in a strip along the exact solution and \( A \) is a linear operator that is the infinitesimal generator of a strongly continuous semigroup \( e^{t A},\) so that exist constants \( M \) and \( w \) such that \( \| e^{t A} \| \leq M \; e^{t w},\) for all \( t \geq 0\). The \(s\)-stage exponential integrators advance the numerical solution of the differential problem from \( ( t_n, u_n \simeq u(t_n) ) \to ( t_{n+1}= t_n + h, u_{n+1})\) by means of the formulae \[ u_{n+1} = u_n + h \; \varphi_1 (h A) \; F (u_n) + h \sum_{i=2}^s b_i (h A) \; D_{n,i} \] with \( D_{n,i} = g( U_{n,i}) - g(u_n)\) \( (i=2, \dots ,s)\) and the stage vectors \( U_{n,i}\) defined recursively by the equations \[ U_{n,i} = u_n + h \; c_i \; \varphi_1 (h c_i A) \; F (u_n) + h \sum_{j=2}^{i-1} a_{ij} ( h A) \; D_{n,j}, (i=2, \dots ,s). \] Here, \( b_i = b_i (Z)\), \( a_{ij} = a_{ij} (Z)\) are matrix-valued linear combinations of the entire functions \( \varphi_k (z)\) with \( \varphi_0 (z) = \exp (z)\) and \( \varphi_{k+1} (z) = z^{-1} ( \varphi_k (z) - \varphi_k (0) ),\) \( k \geq 0\). To get the desired (stiff) order conditions on the available coefficients \( b_i, a_{ij} \) of the above method, the authors introduce an appropriate set of rooted trees and corresponding elementary differentials of the solution \( u(t)\) and the nonlinear term \( g(u)\). This formalism permits the authors to establish the Taylor series expansion of the exact and numerical solutions in powers of the step size \(h\) and then to give the order conditions on the coefficients that are listed in a table up to order 5. Finally, it is shown that the same approach can be used also to derive the stiff-order conditions for exponential Rosenbrock methods.