  A quasiconformal tree is a doubling metric tree in which the diameter of each
arc is bounded above by a fixed multiple of the distance between its endpoints.
We study the geometry of these trees in two directions. First, we construct a
catalog of metric trees in a purely combinatorial way, and show that every
quasiconformal tree is bi-Lipschitz equivalent to one of the trees in our
catalog. This is inspired by results of Herron-Meyer and Rohde for quasi-arcs.
Second, we show that a quasiconformal tree bi-Lipschitz embeds in a Euclidean
space if and only if its set of leaves admits such an embedding. In particular,
all quasi-arcs bi-Lipschitz embed into some Euclidean space.
