In this work, we study the composition operators on the little Lipschitz space \({\mathcal {L}}_0\) of a rooted tree T, defined as the subspace of the Lipschitz space \({\mathcal {L}}\) consisting of the complex-valued functions f on T such that \(\lim _{|v|\rightarrow \infty }|f(v)-f(v^-)|=0,\) where \(v^-\) is the vertex adjacent to the vertex v in the path from the root to v and |v| denotes the number of edges from the root to v. Specifically, we give a complete characterization of the self-maps \(\varphi \) of T for which the composition operator \(C_\varphi \) is bounded and we estimate its operator norm. In addition, we study the spectrum of \(C_\varphi \) and the hypercyclicity of the operators \(\lambda C_\varphi \) for \(\lambda \in {\mathbb {C}}\) .
