"The Anniversary Song" is the first theme of a four-part Viennese-style waltz by the Romanian composer Ion (or Iosif) Ivonovici called "Waves of the Danube" and published in 1880. Al Jolson and Saul Chaplin popularised "The Anniversary Song" in the USA with their adaptation of this theme in 1946. The other three themes in "Waves of the Danube" are pleasant but - to my ear - don't have the same resonance as the first part.

And if you want to hear it played on tenor guitar, mandolin, guitar and acoustic bass, here it is:

The Anniversary Song