This paper is a continuation of the analysis of Edgeworth expansions for one-split branching random walk and its application to random trees. We provide new results for profile, mode and width for several simply generated random trees, in particular forrandom recursive trees, p-oriented recursive trees and D-ary random trees. Our results are corollaries of a general Edgeworth expansion for a one-split branching random walk proved by Kabluchko, Marynych and Sulzbach [The Annals of Applied Probability 27(6): 3478-3524, 2017]. We derive an additional characterization of the random variables appearing in the coefficients of the asymptotic expansions by calculating explicitly corresponding fixed-point equations of a branching type. We further provide numerical simulations justifying our theoretical findings.
