We study coagulation-fragmentation equations inspired by a simple model derived in fisheries science to explain data on the size distribution of schools of pelagic fish. The equations lack detailed balance and admit no H-theorem, but we are anyway able to develop a rather complete description of equilibrium profiles and large-time behavior, based on complex function theory for Bernstein and Pick (Herglotz) functions. The generating function for discrete equilibrium profiles also generates the Fuss-Catalan numbers that count all ternary trees with nn nodes. The structure of equilibrium profiles and other related sequences is explainedthrough a new and elegant characterization of the generating functions of completely monotone sequences, as those Pick functions analytic and nonnegative on (−∞,1)(−∞,1). Self-similar solutions with infinite first moment exist that attract data with the same initial tail behavior.