Title: Shapes of trees
Abstract: A finite tree in the plane is *conformally balanced* if every edge has the same harmonic measure as seen from infinity, and harmonic measures on the two sides of every edge are identical. It is well known that a finite tree has a conformally balanced shape, which is unique up to scale. In this talk, we study shapes of infinite trees, focusing on the case of an infinite trivalent tree. To conformally balance the infinite trivalent tree, we truncate it at level n, form the true tree T_n and take n to infinity. We show that the Hausdorff limit of the T_n contains the boundary of the developed deltoid, the domain obtained by repeatedly reflecting the deltoid in its sides. This is joint work with P. Lin, S. Rohde and E. Sygal.