That's not how religion, culture or traditions work. That there wasn't a tree at the nativity scene doesn't mean a thing - the nativity scene didn't appear out of thin air when Christianity was formed either - it appeared centuries after.

The gifts, tree and all the rest are culturally Christian traditions, just like eating specific foods or giving Hanukkah money is Jewish cultural tradition. It cannot be divorced from it, and it is not some universal tradition with no background. It doesn't make it bad or anything, but it's important to acknowledge it for what it is.