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Counting leaf-labeled rooted binary trees with n distinct labeled leaves is a classical problem that admits a simple closed-form solution. In contrast, when the labels are removed, no simple closed-form expression is known. Despite this difference, the two problems exhibit similar levels of complexity from an asymptotic point of view. In this talk, we explain this phenomenon and present asymptotic counting results that apply in both settings. If time permits, we also discuss several generalizations, including unrooted trees, leaf-multilabeled trees (where labels may repeat), and non-binary trees.

More information: https://web.math.nccu.edu.tw/mfuchs/