mountain ash vs eucalyptus

mountain ash and eucalyptus both are nouns.

Word	Noun	Adjective	Verb	Adverb
mountain ash	Yes	No	No	No
eucalyptus	Yes	No	No	No

As nouns, eucalyptus is a hypernym of mountain ash; that is, eucalyptus is a word with a broader meaning than mountain ash:

mountain ash: tree having wood similar to the alpine ash; tallest tree in Australia and tallest hardwood in the world
eucalyptus: a tree of the genus Eucalyptus

Other hypernyms of mountain ash include eucalypt, eucalyptus tree.

mountain ash (noun)	eucalyptus (noun)
any of various trees of the genus Sorbus	a tree of the genus Eucalyptus
low-growing ash of Texas	wood of any of various eucalyptus trees valued as timber
tree having wood similar to the alpine ash; tallest tree in Australia and tallest hardwood in the world