petition vs bespeak

petition is a noun but bespeak is not a noun.

Word	Noun	Adjective	Verb	Adverb
petition	Yes	No	Yes	No
bespeak	No	No	Yes	No

As verbs, bespeak is a hypernym of petition; that is, bespeak is a word with a broader meaning than petition:

petition: write a petition for something to somebody; request formally and in writing
bespeak: express the need or desire for

Other hypernyms of petition include ask for, call for, quest, request.

petition (noun)	bespeak (noun)
a formal message requesting something that is submitted to an authority
reverent petition to a deity

petition (verb)	bespeak (verb)
write a petition for something to somebody; request formally and in writing	express the need or desire for
	be a signal for or a symptom of