The height of discrete-time critical beta-splitting trees

We determine the first-order asymptotic height of the discrete-time critical beta-splitting tree. If $L_n^{\ast }$ denotes the height (maximum root-to-leaf graph distance) of the tree with $n$ leaves, then $$\frac{L_n^{\ast }}{(\log n{)}^2}⟶C_{\mathrm{h}\mathrm{t}}:=\underset{\theta >1}{\min }\frac{\theta }{2\{\psi (\theta )+\gamma \}}\approx 0.976$$ almost surely and in $L^p$ for every fixed $p>1$ as $n\to \mathrm{\infty }$. Here $\psi$ is the digamma function and $\gamma$ is Euler’s constant. This answers \cite[Open Problem~4]{AldousJansonII} of Aldous and Janson~.