The lyrics have been posted in these threads:

old apple tree in the orchard (version #1)

THE OLD APPLE TREE (version #2)

The song "was written by M.K. Jerome and Jack Scholl for the Humphrey Bogart movie, 'Swing Your Lady' (1938), and the tune was subsequently featured as background music to several Warner Brothers cartoons. Bob Crosby and His Orchestra also recorded a version of it." (From HERE).
The Old Apple Tree [Realaudio] sung by A'mt Idy Harper & Coon Creek Girls (Vocalion 04203, June 1938), from Honkingduck.

~Masato