blue gum vs eucalyptus tree

blue gum and eucalyptus tree both are nouns.

Word	Noun	Adjective	Verb	Adverb
blue gum	Yes	No	No	No
eucalyptus tree	Yes	No	No	No

As nouns, eucalyptus tree is a hypernym of blue gum; that is, eucalyptus tree is a word with a broader meaning than blue gum:

blue gum: tall fast-growing timber tree with leaves containing a medicinal oil; young leaves are bluish
eucalyptus tree: a tree of the genus Eucalyptus

Other hypernyms of blue gum include eucalypt, eucalyptus.

blue gum (noun)	eucalyptus tree (noun)
tall fast-growing timber tree with leaves containing a medicinal oil; young leaves are bluish	a tree of the genus Eucalyptus