  We study Schr\"odinger operators on trees and construct associated Poisson
kernels, in analogy to the laplacian on the unit disc. We show that in the
absolutely continuous spectrum, the generalized eigenfunctions of the operator
are generated by the Poisson kernel. We use this to define a "Fourier
transform", giving a Fourier inversion formula and a Plancherel formula, where
the domain of integration runs over the energy parameter and the geometric
boundary of the tree.
