The diffusion-convection transport process in a:branching fin with a power-law type transfer coefficient (h = a Theta(m) was investigated theoretically. The governing equations were formulated and the solution was obtained analytically. For systems with m>-1, the steady-state distribution for the state variable Theta was unique and stable. When the exponent becomes less than -1, nevertheless, bistability occurs if the steady-state solution exists. Linear stability analysis shows that the lower branch solution on the fin efficiency/effectiveness versus branching number plot was stable, whereas the upper solution was unstable to small perturbations.