On the Number of Prefix and Border Tables

For some text algorithms, the real measure for the complexity analysis is not the string itself but its structure stored in its prefix table (or border table, as border and prefix tables can be proved to be equivalent). We give a new upper bound on the number of prefix tables for strings of length n (on any alphabet) which is of order (1 + φ) (with φ = 1+ √ 5 2 the golden mean) and present also a lower bound.