On growing random binary trees

A sequence {T_n}_{n = 1}^∞ of nested binary trees generated by an infinite sequence of i.i.d. random variables is studied. Two absolute constants β₁,β₂ are shown to exist (0.37 < β₁ < 0.50, 3.58 < β₂ < 4.32), such that $\text{lim}\text{h}{}_{\mathrm{n}}\text{ln}\text{n = β}{}_1\text{,}\text{lim}\text{H}{}_{\mathrm{n}}\text{/}\text{ln}\text{n = β}{}_2$ with probability one; here h_n and H_n are respectively the lengths of the shortest and the longest branches of the tree T_n.