cut down vs shave

cut down is not a noun while shave is a noun.

Word	Noun	Adjective	Verb	Adverb
cut down	No	No	Yes	No
shave	Yes	No	Yes	No

As verbs, shave is a hyponym of cut down; that is, shave is a word with a more specific, narrower meaning than cut down:

cut down (noun)	shave (noun)
	the act of removing hair with a razor

cut down (verb)	shave (verb)
cut down on; make a reduction in	remove body hair with a razor
intercept (a player)	touch the surface of lightly
cause to come or go down	cut closely
cut with a blade or mower	cut or remove with or as if with a plane
cause to fall by or as if by delivering a blow	make shavings of or reduce to shavings
cut with sweeping strokes; as with an ax or machete	cut the price of