  We consider a generalised version of Motzkin paths, where horizontal steps
have length $\ell$, with $\ell$ being a fixed positive integer. We first give
the general functional equation for the area-length generating function of this
model. Using a heuristic ansatz, we derive the area-length scaling behaviour in
terms of a scaling function in one variable for the special cases of Dyck,
(standard) Motzkin and Schr\"oder paths, before generalising our approach to
arbitrary $\ell$. We then derive an expression for the generating function of
Schr\"oder paths and analyse the scaling behaviour of this function rigorously
in the vicinity of the tri-critical point of the model by applying the method
of steepest descents for the case of two coalescing saddle points. Our results
show that for Dyck and Schr\"oder paths, the heuristic scaling ansatz
reproduces the rigorous results.
