Diffusion-limited aggregation on a tree

We study the following growth model on a regular d-ary tree. Points at distance n adjacent to the existing subtree are added with probabilities proportional to alpha(-n), where alpha < 1 is a positive real parameter. The heights of these clusters are shown to increase linearly with their total size; this complements known results that show the height increases only logarithmically when alpha greater than or equal to 1. Results are obtained using stochastic monotonicity and regeneration results which may be of independent interest. Our motivation comes from two other ways in which the model may be viewed: as a problem in fist-passage percolation, and as a version of diffusion-limited aggregation (DLA), adjusted so that ''fingering'' occurs.