Jacobsthal Trees and Generalized κ · x ± 1 Transformations

We introduce a structural framework for the study of generalizedκ x ± 1 transformations on the set of natural numbers.The approach is based on generalized Jacobsthal sequences,which generate a family of branching structures called Jacobsthal trees.Within this framework, classical Collatz-type iterationsappear as reverse trajectories along the branches of these trees.We establish periodicity properties of the node structure,derive recurrence relations governing the formation of branching nodes,and prove structural identities linking consecutive periods.These results provide a structural reformulation ofκ x ± 1 discrete dynamical systemsin terms of Jacobsthal-type constructions.The proposed representation reveals intrinsic periodic symmetriesand partition properties of the set Nthat are not explicit in the standard iterative formulation.