The "true love" mentioned in the song doesn't refer to an earthly suitor, it refers to God Himself.
For Catholics and Christians Christmas starts on Christmas Day and lasts through the feast of the Epiphany, when the Magi brought gifts to the Christ Child.
So Catholics and many Christians put up their Christmas trees on Christmas eve and while the unbelievers take down their trees, these people leave theirs up through January 6th.