"Don't sound like folk to me, except maybe bluegrass."

Yet there's a million different versions of every song, as interpreted by different singers and instrumentalists.

This can be described as filling in the flesh onto a skeleton structure, which is basically how Jazz works.

Remember that improvisation doesn't have to mean solo's, it can also mean harmonization, comping, rhythmic flexibility and reharmonization (using different chords).

The thing that Jazz and folk have in common, unlike classical music, is that specific parts are not preordained for different instruments.

In fact, specific instruments are not ... erm ... specified.

She moved through the fair can be sung unaccompanied, or it can be played on the pipes with a harp accompanist or any other way you choose to play it if you lke it.

The important thing is the melody, so you are free to reharmonize or improvise youe own accompaniment. You are also free to embellish the tune as you wish.

Taste is the arbiter of the extent to which a melody may be improvised upon ... just as one would find in Jazz ...

African folk and African American folk (blues) are nothing if not centred around free interpretation and improvisation. That is the main source from which Jazz got the idea, though not the sole source.

I don't know much about bluegrass so I can't really comment on that comparison.