  Using uniformization, Cantor type sets can be regarded as boundaries of
rooted trees. In this setting, we show that the trace of a first-order Sobolev
space on the boundary of a regular rooted tree is exactly a Besov space with an
explicit smoothness exponent. Further, we study quasisymmetries between the
boundaries of two trees, and show that they have rough quasiisometric
extensions to the trees. Conversely, we show that every rough quasiisometry
between two trees extends as a quasisymmetry between their boundaries. In both
directions we give sharp estimates for the involved constants. We use this to
obtain quasisymmetric invariance of certain Besov spaces of functions on Cantor
type sets.
