Year: 2026
Type: preprint
Abstract: We study the extreme local structure of plane binary trees through the limiting distribution of the number of leaves at maximum depth. For every $m\geq1$, we determine the limiting probability $κ^{[m]}$ that the deepest level contains $2m$ leaves. The resulting sequence is encoded by a generating function $K(y)$, obtained from a bivariate refinement of the Catalan iteration. We show that, at the critical point of this iteration, a parabolic recurrence governs the marked generating functions. Using continuous iteration theory and a Fatou coordinate, we transform this recurrence into the functional equation $K((1+y/4)^2)=K(y)+4$, which describes the limiting law. Its singular behavior yields the exponential tail $κ^{[m]}\sim4^{-m+1}$. Numerical evaluation gives, e.g., $κ^{[1]}\approx0.7009$, $κ^{[2]}\approx0.2226$ and an average number of deepest leaves equal to $\hatκ=4κ^{[1]}\approx2.8037$.
Source: arXiv (Cornell University)
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Agence Nationale de la Recherche
OpenAlex ID: W7161180351