We consider an island growth mechanism wherein species adsorbed on top of two-dimensional islands diffuse to the edge and are then incorporated. A simple matrix formulation is developed for this random-walk problem which allows exact calculation of the growth probability distribution, neglecting island rearrangement. Furthermore, we also show how related trapping probabilities can be calculated recursively for various stages of island growth. The case of imperfect trapping at island edges is also considered, and reduction to Eden-model–type growth in the low-trapping-probability regime is demonstrated. In general, the growing islands are rounder and have fewer defects and narrower active zones than Eden clusters.