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"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 rutigum (nominative rugitus) "a bellowing," from past participle of Latin rugire "to bellow," from PIE imitative root *reu-. The verb is recorded from early 15c. Related: Rutting.