2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60490We characterize the ``best'' model geometries for the class of virtually free groups, and we show that there is a countable infinity of distinct ``best'' model geometries in an appropriate sense--these are the maximally symmetric trees. The first theorem gives several equivalent conditions on a bounded valence, cocompact tree T without valence 1 vertices saying that T is maximally symmetric. The second theorem gives general constructions for maximally symmetric trees, showing for instance that every virtually free group has a maximally symmetric tree for a model geometry.37 pages. A minor revision, correcting a few typos, grammar errors, and omissionsGroup Theory20F65text