cut down vs push down

cut down and push down both are verbs.

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

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

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

cut down (verb)	push down (verb)
cut down on; make a reduction in	cause to come or go down
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