We are interested in the independence number of large random simply generated trees and related parameters, such as their matching number or the kernel dimension of their adjacency matrix. We express these quantities using a canonical tricolouration, which is a way to colour the vertices of a tree with three colours. As an application we obtain limit theorems in L^p for the renormalised independence number in large simply generated trees (including large size-conditioned Bienaymé-Galton-Watson trees).