  We propose here a general framework to address the question of trace
operators on a dyadic tree. This work is motivated by the modeling of the human
bronchial tree which, thanks to its regularity, can be extrapolated in a
natural way to an infinite resistive tree. The space of pressure fields at
bifurcation nodes of this infinite tree can be endowed with a Sobolev space
structure, with a semi-norm which measures the instantaneous rate of dissipated
energy. We aim at describing the behaviour of finite energy pressure fields
near the end. The core of the present approach is an identification of the set
of ends with the ring Z_2 of 2-adic integers. Sobolev spaces over Z_2 can be
defined in a very natural way by means of Fourier transform, which allows us to
establish precised trace theorems which are formally quite similar to those in
standard Sobolev spaces, with a Sobolev regularity which depends on the growth
rate of resistances, i.e. on geometrical properties of the tree. Furthermore,
we exhibit an explicit expression of the "ventilation operator", which maps
pressure fields at the end of the tree onto fluxes, in the form of a
convolution by a Riesz kernel based on the 2-adic distance.
