The height of discrete-time critical beta-splitting trees

We determine the asymptotic height of the discrete-time critical beta-splitting tree. Let $L_n^*$ denotes the height of the tree with $n$ leaves, defined as the maximum graph distance from the root to a leaf. Then, \[ \frac{L_n^*}{(\log n)^2} \longrightarrow C_{\mathrm{ht}}:=\min_{θ>1} \frac