A decreasing rearrangement for functions on homogeneous trees

We introduce a new decreasing rearrangement for functions defined on a homogeneous tree, which enjoys very intuitive and natural properties. The idea is to iterate, with respect to a particular ordering, the usual one dimensional rearrangement on each geodesic. After showing the canonicity of this definition and the axioms of symmetrization, we prove our main result: the geometric and analytic definitions, in terms of the “layer cake” formula, agree.