A finite planar tree has many topologically equivalent drawings in the plane; is there a most natural way to draw it? One possible choice is called the "true form" of the tree. It arises from algebraic geometry and is closely related to Grothendieck's dessins d'enfants. I will describe the true form of a tree (a true tree) in different terms, using harmonic measure, Brownian motion, and conformal maps, and then prove that every planar tree has a true form by using the measurable Riemann mapping theorem. I will then discuss the possible shapes of true trees, e.g., can any compact connected set can be approximated by true trees? If time permits, I will mention the analogous problem for infinite planar trees and some applications to holomorphic dynamics.
This talk is more analysis than discrete geometry, but I will try to explain or avoid the technicalities.