"narrow track worn or cut in the ground," 1570s, probably from Middle English route (see route (n.)); though OED finds this "improbable." Metaphoric meaning "narrow, monotonous routine; habitual mode of behavior" first attested 1839.
"annually recurring sexual excitement in animals; animal mating season" (originally of deer), early 15c., from Old French rut, ruit, from Late Latin rugitum (nominative rugitus) "a bellowing, a roaring," from past participle of Latin rugire "to bellow," from PIE imitative root *reu-. The verb is recorded from early 15c. Related: Rutting.