Hypercyclic algebras for weighted shifts on trees

We study the existence of algebras of hypercyclic vectors for weighted backward shifts on sequence spaces of directed trees with the coordinatewise product. When $V$ is a rooted directed tree, we show the set of hypercyclic vectors of any backward weighted shift operator on the space $c_0(V)$ or $\ell^1(V)$ is algebrable whenever it is not empty. We provide necessary and sufficient conditions for the existence of these structures on $\ell^p(V), 1<p<+\infty$. Examples of hypercyclic operators not having a hypercyclic algebra are found. We also study the existence of mixing and non-mixing backward weighted shift operators on any rooted directed tree, with or without hypercyclic algebras. The case of unrooted trees is also studied.