star anise vs anise tree

star anise and anise tree both are nouns.

Word	Noun	Adjective	Verb	Adverb
star anise	Yes	No	No	No
anise tree	Yes	No	No	No

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

star anise: small tree of China and Vietnam bearing anise-scented star-shaped fruit used in food and medicinally as a carminative
anise tree: any of several evergreen shrubs and small trees of the genus Illicium

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

star anise: small shrubby tree of Japan and Taiwan; flowers are not fragrant
anise tree: any of several evergreen shrubs and small trees of the genus Illicium

star anise (noun)	anise tree (noun)
anise-scented star-shaped fruit or seed used in Asian cooking and medicine	any of several evergreen shrubs and small trees of the genus Illicium
small tree of China and Vietnam bearing anise-scented star-shaped fruit used in food and medicinally as a carminative
small shrubby tree of Japan and Taiwan; flowers are not fragrant