Fractal functions with no radial limits in Bergman spaces on trees

For each p > 0 we provide the construction of a harmonic function on a homogeneous isotropic tree T in the Bergman space A(p) (sigma) with no finite radial limits anywhere. Here, sigma is an analogue of the Lebesgue measure on the tree. With the appropriate modifications, the construction yields a function in A(1) (sigma) when T is a rooted radial tree such that the number of forward neighbors increases so slowly that their reciprocals are not summable.