  Several authors have studied convergence in distribution to the Brownian web
under diffusive scaling of Markovian random walks. In a paper by R. Roy, K.
Saha and A. Sarkar, convergence to the Brownian web is proved for a system of
coalescing random paths -- the Random Directed Forest -- which are not
Markovian. Paths in the Random Directed Forest do not cross each other before
coalescence. Here we study a generalization of the non-Markovian Random
Directed Forest where paths can cross each other and prove convergence to the
Brownian web. This provides an example of how the techniques to prove
convergence to the Brownian web for systems allowing crossings can be applied
to non-Markovian systems.
