  In this paper we start a systematic study of quantum field theory on random
trees. Using precise probability estimates on their Galton-Watson branches and
a multiscale analysis, we establish the general power counting of averaged
Feynman amplitudes and check that they behave indeed as living on an effective
space of dimension 4/3, the spectral dimension of random trees. In the `just
renormalizable' case we prove convergence of the averaged amplitude of any
completely convergent graph, and establish the basic localization and
subtraction estimates required for perturbative renormalization. Possible
consequences for an SYK-like model on random trees are briefly discussed.
