The main objective of the study is to introduce branches of Sheffer stroke BCK-algebras due their specific elements. At the onset of the study, an atom of a Sheffer stroke BCK-algebra is defined and it is shown that the set of all atoms of the algebraic structure is its subalgebra. Then it is proved that specified subsets defined by atoms of a Sheffer stroke BCK-algebra are ideals but the inverses are not true in general. Moreover, a branch and a chain on a Sheffer stroke BCK-algebra are introduced and some properties are presented. Finally, relationships between aforementioned concepts are built and supported by illustrative examples.