The paper develops a variation on Context Tree Weighting (CTW) which keeps memory costs low by adapting the depth of each branch to the extent that it aids prediction accuracy. The new algorithm, called Utile Context Tree Weighting (UCTW), is shown empirically in some illustrative examples to use less memory than fixed-depth CTW (since it can keep some branches short) and to be more effective under a memory bound (in which it must prune a node every time it expands a node). The experiments are, for the most part well designed to answer the questions being asked. One experiment that felt less well-posed was the T-Maze. The text says "We consider a maze of length 4. Thus we set K 3." What does that "thus" mean?