One-sided Variations on Tries: Path Imbalance, Climbing, and Key Sampling

One-sided variations on path length in a trie (a sort of digital trees) are investigated: They include imbalance factors, climbing under different strategies, and key sampling. For the imbalance factor accurate asymptotics for the mean are derived for a randomly chosen key in the trie via poissonization and the Mellin transform, and the inverse of the two operations. It is also shown from an analysis of the moving poles of the Mellin transform of the poissonized moment generating function that the imbalance factor (under appropriate centering and scaling) follows a Gaussian limit law. The method extends to several variations of sampling keys from a trie and we sketch results of climbing under different strategies. The exact probability distribution is computed in one case, to demonstrate that such calculations can be done, at least in principle.