The infinite trivalent tree and the developed deltoid

Every finite tree has an essentially unique conformally natural embedding into the complex plane, by means of "Shabat polynomials". How about infinite trees? Numerical computations showed that large trivalent trees approximate the "developed deltoid", a fascinating domain defined through repeated reflection of the deltoid curve over its three sides, introduced and studied by Lee, Ljubich, Makarov and Mukherjee.
In the first part of my talk, I will carefully describe these objects and discuss a form of convergence of the trees to the developed deltoid. In the second part, I will discuss some techniques used in the proof, and will hint at connections to random structures (the continuum random tree and the Peano sphere).