What is the universal cover of the plane minus two points?

Question

I know that the universal cover of the plane minus the origin is the plane with the exponential map, but I can't think of the analogue with two points removed. I figured out what the universal cover of two wedged circles looks like (a sort of fractal-like infinite tree thingy in the plane), but I'm not sure if this is relevant (my guess that it might be relevant is that the two are homotopic).

Answer 1

The universal cover of the plane minus two points is a plane. The covering map, though, is complicated!