In a classical Markov chain with finite state space S, the law of the process evolves linearly via the transition matrix. The trajectory of the chain can be seen as generating a random labelling of the infinite path. If one instead wishes to label an infinite m-ary tree, the law at each vertex is given by a multilinear function of the laws of its parents. We show an ergodicity condition under which this process has a unique invariant law and converges to it geometrically.