A simple analysis is given for the coupling of a pion to baryons of arbitrary spins and parities. The structure of the vertex depends on a new quantum number called $\ensuremath{\gamma}$ parity. Recursion formulas are given relating high-spin vertices to lower spin vertices. The recursion formulas are solved to give explicit formulas for the $\ensuremath{\pi}B{B}^{\ensuremath{'}}$ vertex when one baryon is at rest. New polynomials in the energy variable are introduced for this purpose. A symmetry relation is derived which connects vertex functions of opposite $\ensuremath{\gamma}$ parity and opposite energy of the moving baryon. The isospin structure of the vertex is analyzed in order to obtain a standardized form useful in comparing effective coupling constants. "Transition isospin matrices" are introduced and their properties derived. General, explicit formulas for decay widths are given for arbitrary-spin transitions of the baryons. No trace calculations are required.