Tight Upper and Lower Bounds on the Path Length of Binary Trees

Abstract

The external path length of a treeT is the sum of the lengths of the paths from the root to each external node. The maximal path length difference, $\Delta $, is the difference between the lengths of the longest and shortest such paths Tight lower and upper bounds are proved on the external path length of binary trees with N external nodes and maximal path length difference $\Delta $ is prescribed. In particular, an upper bound is given that, for each value of $\Delta $, can be exactly achieved for infinitely many values of N. This improves on the previously known upper bound that could only be achieved up to a factor proportional to N. An elementary proof of the known upper bound is also presented as a preliminary result. Moreover, a lower bound is proved that can be exactly achieved for each value of N and $\Delta \leqslant {N / 2}$.