We present an analytic solution for growth of branched aggregates or polymers with distributed cluster (dendrimer) size. Monomer addition to each branch follows first-order polymerization kinetics leading to a distribution of branch lengths. The rate constant for monomer addition is considered diffusion-dependent. Deterministic branching occurs so that at prescribed times t(j) (j greater than or equal to 1), p branches emanate from the tip of a branch that began to grow at t(j-1). When the ratio of average branch lengths is constant, L-j/Lj-1 = a, the fractal dimension of branches is In(p)/\ln(a)\. Closed expressions for cluster mass moments show unbounded growth with time unless ap < 1. The number of clusters (zeroth moment) is constant during growth and equal to the number of initiating buds. Expressions for the number of branches and clusters, average cluster mass, volume, density, and viscosity in solution are functions of p and a. Density and molecular weight show features similar to observed behavior of dendrimers and hyper-branched polymers.