This is some 1920s novelty song (if you couldn't tell that by the title), and all the period versions on youtube are sort of charmingly old but have these terrible chirping sound effects. (Given the state of recording at the time, does that mean that someone was poking at some parakeet through little bars at the appropriate beats? Sawdust on the floor. A man has a microphone.)
The song is given a haunting quality which has stayed with me in Francis Ford Coppola's 1974 film The Conversation. Gene Hackman is a surveillance man, wiretapping and spying on a couple's discussion and returning to try to solve its clues throughout the movie. The girl half-sings half-recites this song, which becomes a sad repeated symbol of innocence. It's a beautiful, sad, and paranoid movie, and much more resonant than the Godfathers which bookended it. There's also one impossibly terrifying scene, and John Cazale, Robert Duvall, and Harrison Ford.
I couldn't find that version, or any sad one, but I did find a couple nice jazzy takes by Doris Day and Louis Armstrong.