  We consider $t$-Lee-error-correcting codes of length $n$ over the residue
ring $\mathbb{Z}_m := \mathbb{Z}/m\mathbb{Z}$ and determine upper and lower
bounds on the number of $t$-Lee-error-correcting codes. We use two different
methods, namely estimating isolated nodes on bipartite graphs and the graph
container method. The former gives density results for codes of fixed size and
the latter for any size. This confirms some recent density results for linear
Lee metric codes and provides new density results for nonlinear codes. To apply
a variant of the graph container algorithm we also investigate some geometrical
properties of the balls in the Lee metric.
