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We prove that a new, irreversible growth algorithm, Non-Deletion Reaction- Limited Cluster-cluster Aggregation (NDRLCA), produces equilibrium Branched Polymers, expected to exhibit Lattice Animal statistics [1]. We implement NDRLCA, off-lattice, as a computer simulation for embedding dimension d = 2 and 3, obtaining values for critical exponents, fractal dimension D and cluster mass distribution exponent τ : d = 2, D 1.53 ± 0.05, τ = 1.09 ± 0.06 ; d = 3, D = 1.96 ± 0.04, τ = 1.50 ± 0.04 in good agreement with theoretical LA values. The simulation results do not support recent suggestions [2] that BPs may be in the same universality class as percolation. We also obtain values for a model-dependent critical fugacity, z c and investigate the finite-size effects of our simulation, quantifying notions of inbreeding that occur in this algorithm. Finally we use an extension of the NDRLCA proof to show that standard Reaction-Limited Cluster-cluster Aggregation is very unlikely to be in the same universality class as Branched Polymers/Lattice Animals unless the backbone dimension for the latter is considerably less than the published value.