  We describe both the Bunce-Deddens C*-algebras and their Toeplitz versions,
as crossed products of commutative C*-algebras by partial automorphisms. In the
latter case, the commutative algebra has, as its spectrum, the union of the
Cantor set and a copy the set of natural numbers N, fitted together in such a
way that N is an open dense subset. The partial automorphism is induced by a
map that acts like the odometer map on the Cantor set while being the
translation by one on N. From this we deduce, by taking quotients, that the
Bunce-Deddens C*-algebras are isomorphic to the (classical) crossed product of
the algebra of continuous functions on the Cantor set by the odometer map.
