sweet chestnut vs chestnut tree

sweet chestnut and chestnut tree both are nouns.

Word	Noun	Adjective	Verb	Adverb
sweet chestnut	Yes	No	No	No
chestnut tree	Yes	No	No	No

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

sweet chestnut: wild or cultivated throughout southern Europe, northwestern Africa and southwestern Asia
chestnut tree: any of several attractive deciduous trees yellow-brown in autumn; yield a hard wood and edible nuts in a prickly bur

Other hypernyms of sweet chestnut include chestnut.

sweet chestnut (noun)	chestnut tree (noun)
wild or cultivated throughout southern Europe, northwestern Africa and southwestern Asia	any of several attractive deciduous trees yellow-brown in autumn; yield a hard wood and edible nuts in a prickly bur