  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>=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 first-passage percolation, and as a version of diffusion-limited
aggregation (DLA), adjusted so that `fingering' occurs.
