Using the recently proposed real space renormalisation group method for directed systems, the authors study the critical behaviour of directed self-avoiding walks (DSAW) on both directed lattice animals (DLA) and directed percolation clusters at threshold p_c (DPC) in two dimensions. The values for the exponent nu _{perpendicular to} are found to be nu _{perpendicular to DSAW}^DLA approximately=0.590 and nu _{perpendicular to DSAW}^DPC approximately=0.528, both of which are higher than the mean-field value 0.5. It is also shown rigorously that nu /sub /// remains unchanged; i.e. nu /sub //DSAW/^DLA= nu /sub //DSAW/^DPC=1.