The generalized Fibonacci recurrence gn=gn−k+gn−m was recently used to demonstrate the theoretically optimal nature of limited senescence in morphologically symmetrically dividing bacteria. Here, we study this recurrence from a more abstract viewpoint, as a general model for asymmetric branching, and interpret solutions for different initial conditions in terms of branching-related quantities. We provide a compact diagrammatic representation for the evolution of this process which leads to an explicit binomial identity for the sums of elements lying on the diagonals kx+my=n in Pascal's triangle N0×N0∋(x,y)↦(x+yx), previously sought by Dickinson [Dic50], Raab [Raa63], and Green [Gre68].