We study cellular automata (CA) on the unoriented 𝑘-regular tree 𝑇𝑘, i.e. continuous maps acting on vertex-labelings of 𝑇𝑘 which commute with all automorphisms of the tree.We prove that every CA that is asymptotically nilpotent, meaning every configuration converges to the same constant configuration, is nilpotent, meaning each configuration is mapped to that configuration after finite time.

Keywords: Cellular automata, dynamical systems, Nilpotency, trees, group actions, free groups