When soluble metal salts are placed in a silicate solution, chemical gardens grow. These gardens are treelike structures formed of long, thin, hollow tubes. Here we study one particular case: a calcium nitrate pellet in a solution of sodium trisilicate. We observe that tube growth results from a relaxation oscillation. The average period and the average growth rate are approximately constant for most of the structures growth. The period does fluctuate from cycle to cycle, with the oscillation amplitude proportional to the period. Based on our observations, we develop a model of the relaxation oscillations which calculates the average oscillation period and the average tube radius in terms of fundamental membrane parameters. We also propose a model for the average tube growth rate. Predictions are made for future experiments.