Actually, the story I heard was that the Christmas celebration was moved to December 25th so that it would coincide with a holiday called "the birth of the Invincible Sun God Mithras" (I am not kidding). Mithraism was a major competitor to Christianity in the religion market in Ancient Rome, and the Christian authorities didn't want to be losing the weak-willed that wanted to feast with everyone else.

As far as the Christmas tree goes, it's not worshipped by most Christian families (if yours is different, I imagine that's ok too). A fir tree was used as a metaphor by a leading English or German preacher in the Middle Ages in a sermon regarding the birth of Christ, and after this, the people in the area brought trees (small at first, then larger every year) into their homes to remind them of it. People decorate pretty much everything, so the rich tradition of ornaments was developed.

In other myth, I had thought that the day of Christ's birth would have been in April or May, since this is when the shepherds stay and watch their flocks by night, etc etc. This is only done near lambing time, which is in the spring.

These aren't "pagan" traditions, but developments of minor innovations over a long period of time...just like everything else.