Presented by Dr. Askar Iliasov, University of Zurich. Tree graphs provide an appealing example of a geometry that allows for explicit solutions and lies at the intersection of several areas of physics and mathematics. In this talk, I will discuss three physical phenomena on tree graphs: topological states, multifractality of wave functions in non-Hermitian systems, and boundary superconductivity. In each case, trees exhibit distinctive features absent in Euclidean geometry. Flat bands of Euclidean lattices can persist on decorated covering trees, yet the associated states may acquire a nontrivial topological interpretation. Moreover, unlike in the conventional Euclidean-lattice setting, trees can host topological states in the bulk. Wave functions on trees can also be multifractal even in clean systems, without disorder, where multifractality is driven solely by non-Hermiticity. Finally, boundary superconductivity, identified only recently in the Euclidean setting, is strongly enhanced on trees, where the boundary critical temperature can in some cases exceed the bulk critical temperature by several orders of magnitude.