A family tree is associated with a single person in that tree, usually you (the logged in user). I'd call that the anchor of the tree. Everyone is, in some way, related to that individual (sibling, ancestor, decendent, in-laws, and the families of those).
I can think of a couple of things you can do to have "multiple" trees:
1) Create a tree with "someone" as the anchor, along with "parents", then create "siblings" to the anchor (same parents) and have them be the "anchors" to their own trees. This tree would only grow down (have decendents, not ancestors), and all of your characters would belong to the same tree.
2) Come up with a "root" userid (e.g. twoons), then create new userids for each character (e.g. twoons001, twoons002,...). Each root character would have their own tree.