The melody of this song is fairly ingenious. The chord progression for the verse is F, D minor, C, A minor, and then the second time through it just skips the C chord and goes straight to A minor. A minor is the key, and the melody sticks to A minor pentatonic throughout, but the relationship of the notes to the chords is very unusual.
Over the first F major chord, the melody starts on D and moves down to its lower neighbor, C (the fifth, and a chordal tone). The D, of course, is the upper neighbor of the chordal tone, but it's really emphasized as the first note of the melody and seems to color the chord as an added sixth.
Over the D minor chord, we continue moving around the pentatonic scale, first down to A and then back up to C. Once again, the root of the chord has not been voiced and there is some emphasis on a non-chordal tone (the C). Continuing with pentatonic step-wise motion, the melody then moves back down to A as the first note over the C major chord. A moves down to G and here we have an exact repeat of the note sequence (the sixth of the chord moving to the fifth) that was heard over the F chord previously. Once again, there is a sense of added sixth harmony.
The melody then finishes its descent on E, sung (or whistled*) over the A minor chord. Once again, the root of the chord has not been voiced.
Second time through the chord progression, we start one note higher, on E. Nice octave jump up from the lower E, and voiced over the F major chord, we now have major seventh harmony (again, all accomplished with A minor pentatonic notes). The shortened melody for the consequent lyric line that finishes this part of the verse then concludes by pentatonic step-wise descent to C and then A over the D minor chord (the seventh to the fifth), and then staying on that A to finally have a rooted note, on the very last note of the melody, over the tonic A minor chord.