late 14c., putrefien, "to decompose, rot, decay with a fetid smell," from Old French putréfier and directly from Latin putrefacere "to make rotten," from putrere "to stink" (see putrid) + facere "to make, do" (from PIE root *dhe- "to set, put"). Transitive sense of "cause to decompose or rot" is from early 15c. Related: Putrefied; putrefying.
late 15c., "to decrease," also "to decline, deteriorate, lose strength or excellence," from Anglo-French decair, Old North French decair (Old French decheoir, 12c., Modern French déchoir) "to fall, set (of the sun), weaken, decline, decay," from Vulgar Latin *decadere "to fall off," from de "off" (see de-) + Latin cadere "to fall" (from PIE root *kad- "to fall").
Transitive sense of "cause to deteriorate, cause to become unsound or impaired" is from 1530s. Sense of "decompose, rot" is from 1570s. Related: Decayed; decaying.