A note on the asymptotic behavior of the heights in \(b\)-trees for \(b\) large (Q1578478)

Summary: We study the limiting distribution of the height in a generalized tree in which external nodes are capable to store up to \(b\) items (the so called \(b\)-trees). 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 identify five regions of the height distribution that should be compared to three regions obtained for fixed \(b\). We prove that for most \(n\), the limiting distribution is concentrated at the single point \(k_1=\lfloor \log_2 (n/b)\rfloor +1\) as \(n,b\to \infty\). We observe that this is quite different than the height distribution for fixed \(b\), in which case the limiting distribution is of an extreme value type concentrated around \((1+1/b)\log_2 n\). We derive our results by analytic methods, namely generating functions and the saddle point method. We also present some numerical verification of our results.