In fact it's not even obvious to me that a stack of worlds would "converge": it could hit an attractor with period N where N>1, or do something even more funky.

I'm convinced it would never converge, and even if it did I would expect it to converge on something more interesting and elegant, like a cellular automata. I have no idea what a binary tree system would do unless none of the worlds break the symmetry between A and B. In that case it would behave like a stack, and the story assumes stacks can converge.