  The lace expansion is a powerful tool for analysing the critical behaviour of
self-avoiding walks and percolation. It gives rise to a recursion relation
which we abstract and study using an adaptation of the inductive method
introduced by den Hollander and the authors. We give conditions under which the
solution to the recursion relation behaves as a Gaussian, both in Fourier space
and in terms of a local central limit theorem. These conditions are shown
elsewhere to hold for sufficiently spread-out models of networks of
self-avoiding walks in dimensions $d>4$, and for critical oriented percolation
in dimensions $d+1>5$, providing a unified approach and an essential ingredient
for a detailed analysis of the branching behaviour of these models.
