see vs run down

see is a noun but run down is not a noun.

Word	Noun	Adjective	Verb	Adverb
see	Yes	No	Yes	No
run down	No	No	Yes	No

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

see (noun)	run down (noun)
the seat within a bishop's diocese where his cathedral is located

see (verb)	run down (verb)
deem to be	use up all one's strength and energy and stop working
perceive (an idea or situation) mentally	injure or kill by knocking (someone or something) down and passing over the body, as with a vehicle
get to know or become aware of, usually accidentally	pursue until captured
be careful or certain to do something; make certain of something	move downward
deliberate or decide	examine hastily
make sense of; assign a meaning to	deplete
find out, learn, or determine with certainty, usually by making an inquiry or other effort	trace
match or meet
receive as a specified guest
imagine; conceive of; see in one's mind
come together
conduct someone someplace
perceive by sight or have the power to perceive by sight
perceive or be contemporaneous with
see or watch
see and understand, have a good eye
observe as if with an eye
undergo or live through a difficult experience
observe, check out, and look over carefully or inspect
date regularly; have a steady relationship with
go to see a place, as for entertainment
go to see for professional or business reasons
go to see for a social visit
take charge of or deal with