The Theory of Fringe Analysis and Its Application to 2-3 Trees and B-Trees

A fringe analysis method based on a new way of describing the composition of a fringe in terms of tree collections is presented. It is shown that the derived matrix recurrence relation converges to the solution of a linear system involving the transition matrix, even when the transition matrix has eigenvalues with multiplicity greater than one. As a consequence, bounds and some exact results on the expected number of splits per insertion and on the expected depth of the deepest safe node in 2-3 trees and B-trees, on the expected height of 2-3 trees are obtained, and improvements of the bounds on the expected number of nodes in 2-3 trees are derived also. Bounds and some exact results for 2-3 trees and B-trees using an overflow technique are obtained.