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We prove an asymptotic Edgeworth expansion for the profiles of certain randomtrees including binary search trees, random recursive trees and plane-orientedrandom trees, as the size of the tree goes to infinity. All these models can beseen as special cases of the one-split branching random walk for which we alsoprovide an Edgeworth expansion. These expansions lead to new results on mode,width and occupation numbers of the trees, settling several open problemsraised in Devroye and Hwang [Ann. Appl. Probab. 16(2): 886--918, 2006], Fuchs,Hwang and Neininger [Algorithmica, 46 (3--4): 367--407, 2006], and Drmota andHwang [Adv. in Appl. Probab., 37 (2): 321--341, 2005]. The aforementionedresults are special cases and corollaries of a general theorem: an Edgeworthexpansion for an arbitrary sequence of random or deterministic functions$\mathbb L_n:\mathbb Z\to\mathbb R$ which converges in the mod-$\phi$-sense.Applications to Stirling numbers of the first kind will be given in a separatepaper.

General Edgeworth expansions with applications to profiles of random trees