Algebras sharing the same support $\tau$-tilting poset with tree quiver algebras

Happel and Unger reconstructed hereditary algebras from their posets of tilting modules. Inspired by this result, we try removing the assumption to be hereditary. However, it would be unfortunately fail in general: e.g. every selfinjective algebra has the poset consisting of only one point. Therefore, we should consider a generalization of the Happel-Unger's result for posets of support $\tau$-tilting modules, which contains those of tilting modules. In this paper, we spotlight finite dimensional algebras whose support $\tau$-tilting posets coincide with those of tree quiver algebras and give a full characterization of such algebras.