Complement of a tree in the complex plane

Suppose T is a tree in the complex plane . Consider the complement of the tree in the plane to be $\Omega$ i.e. $\Omega = \mathbb{C} - T$ . I have to show that $\Omega$ is conformally equivalent to the complement of the closed unit disk in the plane .

I was thinking that the complement of the tree in the Riemann sphere is simply connected and hence by Uniformization theorem it is conformally equivalent to the open unit disk and further removing the "$\infty$" from the sphere it becomes conformally equivalent to the open unit disk minus the origin and hence the result follows .

It will be helpful if one points out any mistake in the way of my thinking and provides a more elegant and rigorous proof of the fact .

• You can assume that $T$ is bounded, I guess? If so, yes, your reasoning is correct, assuming you have some characterization of simply connected domains which allows you to conclude that the complement of the tree in the Riemann sphere is simply connected. (E.g., by knowing that a subdomain of the Riemann sphere is simply connected if and only if its complement is connected.) – Lukas Geyer May 14 '18 at 18:41