  A relational characterization of cop-win graphs was provided by Nowakowski
and Winkler in their seminal paper on the game of Cops and Robbers. As a
by-product of that characterization, each cop-win graph is assigned a unique
ordinal, which we refer to as a CR-ordinal. For finite graphs, CR-ordinals
correspond to the length of the game assuming optimal play, with the cop
beginning the game in a least favourable initial position. For infinite graphs,
however, the possible values of CR-ordinals have not been considered in the
literature until the present work. We classify the CR-ordinals of cop-win trees
as either a finite ordinal, or those of the form $\alpha + \omega$, where
$\alpha$ is a limit ordinal. For general infinite cop-win graphs, we provide an
example whose CR-ordinal is not of this form. We finish with some problems on
characterizing the CR-ordinals in the general case of cop-win graphs.
