We show that the space of fully grown n-trees has the homotopy type of a bouquet of spheres of dimension n - 3 and that the character of the representation of Σn + 1 on its only non-trivial reduced homology group is ε · (IndΣn + 1Σn Lien - Lien + 1).
Robinson, A., & Whitehouse, S. (1996). The tree representation of Σn + 1. Journal of Pure and Applied Algebra, 111(1–3), 245–253. https://doi.org/10.1016/0022-4049(95)00116-6