public school vs lycee

public school and lycee both are nouns.

Word	Noun	Adjective	Verb	Adverb
public school	Yes	No	No	No
lycee	Yes	No	No	No

As nouns, lycee is a hypernym of public school; that is, lycee is a word with a broader meaning than public school:

public school: private independent secondary school in Great Britain supported by endowment and tuition
lycee: a school for students intermediate between elementary school and college; usually grades 9 to 12

Other hypernyms of public school include Gymnasium, lyceum, middle school, secondary school.

public school (noun)	lycee (noun)
a tuition free school in the United States supported by taxes and controlled by a school board	a school for students intermediate between elementary school and college; usually grades 9 to 12
private independent secondary school in Great Britain supported by endowment and tuition