Conformal laminations and trees

Conformal maps \$f\$ of the unit disc \$\mathbb D\$ have a continuous extension to the circle if (and only if) the boundary of the image \$f\$(\$\mathbb D\$) is locally connected. This extension induces an equivalence relation on the circle by declaring that \$x\$ ∼ \$y\$ if \$f\$(\$x\$) = \$f\$(\$y\$).  Which equivalence relations on the circle arise in this way?

In the first part of my talk, I will discuss the background and history of this problem and give motivation from complex dynamics, computational algebra and probability theory.  In the second part, I will present a characterization under the additional assumption that \$f\$(\$\mathbb D\$) is a John domain whose complement has empty interior.