The tune is Welsh. It is Nos Galan, which means New Year's Eve, a melody long sung in Wales with Welsh words to see the old year out. The English Christmas words, with their boughs of holly and Yuletide carol, came in the 19th century and are usually credited to the Scottish songwriter Thomas Oliphant.
Fa la la has no meaning: it is a refrain sung for its sound, like the nonsense refrains that were already common in English part-songs of the 16th century. In Deck the Halls it carries half the song, since every line of words is answered by a line of fa la las, and that is a large part of why the carol is so easy to sing in a group.
This transcription: Welsh New Year's Eve tune "Nos Galan", as it is sung today with the English Christmas words: the whole verse in C major; left hand by GOONS Studio.
Transcribed by GOONS Studio from the original work, which is in the public domain.
For kids: Deck the Halls by colors, on a rainbow piano that lights the next key and waits.