The Node Visit Cost of Brother Trees

The performance of a brother search tree can be measured by three basic cost measures: node visit cost, comparison cost, and space cost. The structure of brother trees that are optimal with respect to one of these cost measures is already known, as well as how to construct them in linear time. In this paper we investigate sharp bounds for the range that the node visit cost may take for a given size of the tree. To this end we determine the structure of those brother trees which, for a given size N, have maximal (or pessimal) node visit cost. We derive a tight upper bound for the node visit cost of 1–2 brother trees which together with the lower bound obtained earlier yields the desired range estimation. Furthermore, we obtain that at least 11.6% of the internal nodes of a brother tree of maximal height are unary.