A Note on the Asymptotic Behavior of the Heights in b-Tries for b Large

We study the limiting distribution of the height in a generalized trie in which external nodes are capable to store up to b items (the so called b-tries). We assume that such a tree is built from n random strings (items) generated by an unbiased memoryless source. In this paper, we discuss the case when b and n are both large. We shall identify six natural regions of the height distribution that should be compared to three regions obtained for xed b. We prove that for most n, the limiting distribution is concentrated at the single point k1 = blog2(n=b)c + 1 as n; b ! 1. We observe that this is quite di erent than the height distribution for xed b, in which case the limiting distribution is of an extreme value type concentrated around (1+1=b) log2 n. We derive our results by analytic methods, namely generating functions and the saddle point method. We also present some numerical veri cation of our results. This work was supported by DOE Grant DE-FG02-96ER25168. The work of this author was supported by NSF Grants NCR-9415491 and CCR-9804760, and Purdue Grant GIFG.