  Using the generalized normally ordered form of words in a locally-free group
of $n$ generators, we show that in the limit $n\to\infty$, the partition
function of weighted directed lattice animals on a semi-infinite strip
coincides with the partition function of stationary configurations of the
asymmetric simple exclusion process (ASEP) with arbitrary entry/escape rates
through open boundaries. We relate the features of the ASEP in the different
regimes of the phase diagram to the geometric features of the associated
generalized directed animals by showing the results of numerical simulations.
In particular, we show how the presence of shocks at the first order transition
line translates into the directed animal picture. Using the evolution equation
for generalized, weighted Lukasiewicz paths, we also provide a straightforward
calculation of the known ASEP generating function.
