In this paper we study universal coding problems for the integers, in particular, establish rather tight lower and upper bounds for the Elias omega code and other codes. In these bounds, the so-called log-star function plays a central role. Furthermore, we investigate unbounded search trees induced by these codes, including the Bentley-Yao search tree. We will reveal beautiful recursion structures latent in these search trees as well as in these codes. Finally, we introduce the modified log-star function to reveal the existance of better prefix codes than the Elias omega code and other known codes.
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