The average height of the d ‐th highest leaf of a planted plane tree

It is proved that the d th highest leaf, where all planted plane trees with n nodes are assumed to be equally likely, has an average height \documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt {\pi n} ‐ \frac{1}{2} ‐ \frac{2}{3}\sum\nolimits_{s = 1}^{d ‐ 1} {\left({2/9} \right)^s \left({\begin{array}{*{20}c} {2s + 1} \\ s \\ \end{array}} \right)} + O\left({n^{‐ 1/2 + \varepsilon}} \right) $\end{document} for all ϵ > 0 and n → ∞. This solves a problem left open in one of our previous papers.