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We study the average number A_n per site of the number of differentconfigurations of a branched polymer of n bonds on the Given-Mandelbrot familyof fractals using exact real-space renormalization. Different members of thefamily are characterized by an integer parameter b, b > 1. The fractaldimension varies from $ log_{_2} 3$ to 2 as b is varied from 2 to infinity. Wefind that for all b > 2, A_n varies as $ \lambda^n exp(b n ^{\psi})$, where$\lambda$ and $b$ are some constants, and $ 0 < \psi <1$. We determine theexponent $\psi$, and the size exponent $\nu$ (average diameter of polymervaries as $n^\nu$), exactly for all b > 2. This generalizes the earlier resultsof Knezevic and Vannimenus for b = 3 [Phys. Rev {\bf B 35} (1987) 4988].Comment: 24 pages, 8 figure

Branched polymers on the Given-Mandelbrot family of fractals