The Distribution of the Deepest Leaves in Binary Trees

Abstract

We study the extreme local structure of plane binary trees through the distribution of leaves at maximum depth. We first address two basic questions: (i) the asymptotic probability that exactly two leaves occur at the deepest level, and (ii) the asymptotic mean number of leaves at that level. These problems lead to generating functions coupled with the Catalan iteration $I_{k+1}(z)=1+zI_k(z)^2$ through quasi-logistic recurrences. We show that both associated series have dominant singularity $ρ=1/4$ and admit square-root singular expansions. The singular terms are obtained through a three-zone dominated-convergence analysis of the critical scaling regime of the truncation error. We then extend the framework to derive the full limiting distribution of the number of deepest leaves. Enumerating trees with exactly $2m$ deepest leaves yields a hierarchy of differential equations that reduces to successive polynomial integrations. Encoding these parameters into a bivariate generating function transforms the nonlinear dynamics back into the Catalan recurrence. Using continuous iteration theory and the Fatou coordinate associated with an Abel equation, we obtain a functional equation characterizing the distribution. Finally, singularity analysis implies a strict exponential tail: the probability of having $2m$ deepest leaves satisfies $κ[m]\sim 4^{-m+1}$. Numerical evaluation gives an average number of deepest leaves equal to $\hatκ\approx 2.8037$, while the probability of exactly two deepest leaves is $κ\approx 0.7009$.