<p>The Zygmund space of a tree is the Banach space of complex-valued functions defined on the vertices of a rooted infinite and locally-finite tree such that their second discrete derivative is bounded. In this paper we study the forward and backward shift operators on the Zygmund space of a tree. We show that the forward shift operator is always bounded on the Zygmund space, and we find its norm and spectrum. We give necessary and sufficient conditions for the backward shift operator to be bounded, and give an estimate for its norm. In the case the tree is homogeneous, we give an exact expression for the norm of the backward shift operator and we find its spectrum. The results in the paper also apply verbatim to the little Zygmund space.</p>