  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 Bienaym\'e-Galton-Watson trees).
