take away vs cut down

take away and cut down both are verbs.

Word	Noun	Adjective	Verb	Adverb
take away	No	No	Yes	No
cut down	No	No	Yes	No

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

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

take away (verb)	cut down (verb)
take out or remove	cut down on; make a reduction in
remove something concrete, as by lifting, pushing, or taking off, or remove something abstract	intercept (a player)
get rid of something abstract	cause to come or go down
take away a part from; diminish	cut with a blade or mower
buy and consume food from a restaurant or establishment that sells prepared food	cause to fall by or as if by delivering a blow
remove from a certain place, environment, or mental or emotional state; transport into a new location or state	cut with sweeping strokes; as with an ax or machete
take from a person or place