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, thus covering a wider range of functionals. Our main result is illustrated by two explicit examples: the (logarithm of) the number of matchings, and a functional stemming from a tree reduction process that was studied by Hackl, Heuberger, Kropf, and Prodinger.

Let $S_n=X_1+\dots+X_n$ be a sum of independent symmetric random variables such that $\left|X_i\right|\leq1$. Denote by $W_n=\varepsilon_1+\dots+\varepsilon_n$ a sum of independent random variables such that $\mathbb{P}\left\{\varepsilon_i=\pm1\right\}=1/2$. We prove that $\mathbb{P}\left\{S_n\in A\right\}\leq\mathbb{P}\left\{cW_k\in A\right\}$, where $A$ is either an interval of the form $\left[x,\infty\right)$ or just a single point. The inequality is sharp and the optimal values of $c$ and $k$ are given explicitly. It improves Kwapień's inequality in the case of the Rademacher series. We also provide a new and very short proof of the Littlewood--Offord problem without using Sperner's theorem. Finally, an extension to odd Lipschitz functions is given.