We re-examine a two-dimensional forest-fire model via Monte-Carlo simu
lations and show the existence of two length scales with different cri
tical exponents associated with clusters and with the usual two-point
correlation function of trees. We check resp. improve previously obtai
ned values for other critical exponents and perform a first investigat
ion of the critical behaviour of the slowest relaxational mode. We als
o investigate the possibility of describing the critical point in term
s of a distribution of the global density. We find that some qualitati
ve features such as a temporal oscillation and a power law of the clus
ter-size distribution can nicely be obtained from such a model that di
scards the spatial structure.