mountain ash vs eucalyptus tree

mountain ash and eucalyptus tree both are nouns.

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

As nouns, eucalyptus tree is a hypernym of mountain ash; that is, eucalyptus tree 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 tree: a tree of the genus Eucalyptus

Other hypernyms of mountain ash include eucalypt, eucalyptus.

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