No, "Don't Sit Under The Apple Tree" is not in the public domain. The song was written by songwriters Lew Brown, Charles Tobias, and Sam H. Stept, and is still protected by copyright.
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Sammy Davis Jr. recorded his song 'Please Don't Talk About Me
When I'm Gone' in 1966. The song was written by Sam H. Stept, with
lyrics by Sidney Clare.