This talk will focus on the distributions of various parameters associated with random trees, and on the limit distributions of such parameters. Different families of trees will be considered, such as simply generated (Galton–Watson) trees, Pólya trees and increasing trees. The notion of an additive tree functional will play a major role, as it provides us with a rather general approach to study seemingly unrelated parameters of trees, such as the number of leaves, the multiplicity of eigenvalues, and the cardinality of the automorphism group. General asymptotic schemes allow us to prove that many different tree parameters follow a normal or log-normal limit law.