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

The external path length of a tree $T$ 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 $\Delta\leq N/2$.