Extensions of the definition of a (k)-graceful numbering of a finite graph are presented for countably infinite graphs. Sufficient conditions for existence of such numberings are presented. and, in particular, the class of countably infinite trees is examined in detail. It is shown that all countably infinite tress are k-graceful, but not necessarily . Some number theoretic questions, like those of M. Hall and R. Entringer involving various difference sets obtained from the set of natural numbers, are seen to be particular instances of the problem of gracefully labelling a countably infinite graph.
