Shapes of infinite conformally balanced trees

Numerical experiments by Werness, Lee and the third author suggested that dessin d'enfants associated to large trivalent trees approximate the developed deltoid introduced by Lee, Lyubich, Makarov and Mukherjee. In this paper, we confirm this conjecture. As a side product of our techniques, we give a new proof of a theorem of Bishop which says that ``true trees are dense.'' We also exhibit a sequence of trees whose conformally natural shapes converge to the cauliflower, the Julia set of $z\mapsto z^2+1/4$.