eucalyptus tree vs gum tree

eucalyptus tree and gum tree both are nouns.

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

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

eucalyptus tree: a tree of the genus Eucalyptus
gum tree: any of various trees of the genera Eucalyptus or Liquidambar or Nyssa that are sources of gum

Other hypernyms of eucalyptus tree include gum.

eucalyptus tree (noun)	gum tree (noun)
a tree of the genus Eucalyptus	any of various trees of the genera Eucalyptus or Liquidambar or Nyssa that are sources of gum