Song Name?

Greer plays a song on the piano several times in this picture. Ironically (or not) she also plays the same song on piano in the film 'That Forsyte Woman." Now I am watching the 1958 movie with Andy Griffith "No Time for Sergeants" and the same song is played with a hillbilly twang at the beginning of the film. Thanks!!! PS I am watching TCM "Too Youjg To Kiss" right now starring June Allyson and Van Johnson -- the film opens with this very same song. Is it Chopin? My word, this is really getting to me and hope a classical musician can help me out!