shave vs cut down

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

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

As verbs, cut down is a hypernym of shave; that is, cut down is a word with a broader meaning than shave:

Other hypernyms of shave include bring down, cut, cut back, reduce, trim, trim back, trim down.

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

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