push down vs cut down

push down and cut down both are verbs.

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

As verbs, push down and cut down are synonyms defined as:

Other synonyms of push down include down, knock down, pull down.

push down (verb)	cut down (verb)
cause to come or go down	cut down on; make a reduction in
	intercept (a player)
	cause to come or go down
	cut with a blade or mower
	cause to fall by or as if by delivering a blow
	cut with sweeping strokes; as with an ax or machete