How should a person be added to a family tree that was born out of wedlock, and adopted (more than 100 years ago)

Since it's more than 100 years ago, I would not be too worried about scandals and unease. (More recent connections are sometimes very vexed.)

So, add a child to the known parent (s) -- if you know both parents and they were unmarried, you have a choice, on the relationship page of the profile editing, or in the box when you are adding a profile, to specify "partner" rather than "spouse."

Then, for the child, you can add the adoptive parents -- again, on the relationship page, or when you are adding the parents, you can specify "biological," "adoptive," or "foster."