Many parameters of trees are additive in the sense that they can be computed recursively from the sum of the branches plus a certain toll function. For instance, such parameters occur very frequently in the analysis of divide-and-conquer algorithms. Here we are interested in the situation that the toll function is small (the average over all trees of a given size $n$ decreases exponentially with $n$). We prove a general central limit theorem for random labelled trees and apply it to a number of examples. The main motivation is the study of the number of subtrees in a random labelled tree, but it also applies to classical instances such as the number of leaves.