  An additive functional of a rooted tree is a functional that can be
calculated recursively as the sum of the values of the functional over the
branches, plus a certain toll function. Janson recently proved a central limit
theorem for additive functionals of conditioned Galton-Watson trees under the
assumption that the toll function is local, i.e. only depends on a fixed
neighbourhood of the root. We extend his result to functionals that are "almost
local" in a certain sense, thus covering a wider range of functionals. The
notion of almost local functional intuitively means that the toll function can
be approximated well by considering only a neighbourhood of the root. Our main
result is illustrated by several explicit examples including natural graph
theoretic parameters such as the number of independent sets, the number of
matchings, and the number of dominating sets. We also cover a functional
stemming from a tree reduction process that was studied by Hackl, Heuberger,
Kropf, and Prodinger.
