Let T be a tree on n vertices and with maximum degree ∆. We show that for k ≥ ∆ + 1 the Glauber dynamics for k-edge-colourings of T mixes in polynomial time in n. The bound on the number of colours is best possible as the chain is not even ergodic for k ≤ ∆. Our proof uses a recursive decomposition of the tree into subtrees; we bound the relaxation time of the original tree in terms of the relaxation time of its subtrees using block dynamics and chain comparison techniques. Of independent interest, we also introduce a monotonicity result for Glauber dynamics that simplifies our proof. The Glauber dynamics is a Markov chain over the set of configuration of spin systems in the vertices of a graph. In this paper we consider the discrete-time Metropolis Glauber dynamics and study the particular case of sampling k-colourings of graphs. The Glauber dynamics has attracted interest from many different areas. In statistical physics for example, this dynamics gives a single-update sampler for the Gibbs distribution, with the particular case of k-colourings corresponding to the antiferromagnetic Potts model at zero temperature. In computer science, rapid convergence of the Glauber dynamics gives a fully polynomial-time randomised approximation scheme for the number of k-colourings of a graph, a counting problem that is ♯P-complete. Finally, this dynamics is a popular Monte Carlo method for simulating physical systems as it is both relatively simple and easy to implement. The efficiency of these algorithms depends on the time the chain takes to reach equilibrium, known as the mixing time. This is the main object of study in this paper. A well-known conjecture in this area states that the Glauber dynamics for k-colourings of graphs with n vertices and maximum degree ∆ mixes in polynomial time in n for every k ≥ ∆ + 2. This bound on the number of colours is best possible as the chain might not be ergodic for k = ∆ + 1. Jerrum [16], and independently, Salas and Sokal [25], proved that polynomial mixing happens for k ≥ 2∆. Vigoda [30] improved the bound to k ≥ 11∆/6 by analysing the flip chain. Recently, Chen and Moitra [3], and independently, Delcourt, Perarnau and Postle [4] showed that polynomial mixing holds for k ≥ (11/6 − ǫ)∆, for some small ǫ > 0. This is the best known bound for general graphs. The conjecture has also been studied for several classes of graphs such as graphs with large girth [8, 14], random graphs [10, 23], planar graphs [13] and graphs with bounded tree-width [29]. One of the central topics in the area is the study of Glauber dynamics on trees. Although it is quite simple to sample a uniformly random k-colouring of a tree, Glauber dynamics on trees often shows an extremal behaviour that helps in the understanding of its performance on general graphs.