  Let $T$ be a tree on $n$ vertices and with maximum degree $\Delta$. We show
that for $k\geq \Delta+1$ 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 $k \leq \Delta$. 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.
