Dendrites grow by stochastic branching, elongation, and retraction. A key question is whether such a mechanism is sufficient to form highly branched dendritic morphologies. Alternatively, are signals from other cells or is the topological hierarchy of the growing network necessary for dendrite geometry? To answer these questions, we developed a mean-field model in which branch dynamics is isotropic and homogenous (i.e., no extrinsic instruction) and depends only on the average lengths and densities of branches. Branching is modeled as density-dependent nucleation so there are no tree structures and no network topology. Despite its simplicity, the model predicted several key morphological properties of class IV Drosophila sensory dendrites, including the exponential distribution of branch lengths, the parabolic scaling between dendrite number and length densities, the tight spacing of the dendritic meshwork (which required minimal total branch length), and the radial orientation of branches. Stochastic growth also accelerated the overall expansion rate of the arbor. Therefore, stochastic dynamics is an economical and rapid space-filling mechanism for building dendritic arbors without external guidance or hierarchical branching mechanisms. Our model provides a general theoretical framework for understanding how macroscopic branching patterns emerge from microscopic dynamics.