Appearance of vertices of infinite order in a model of random trees

We study an equilibrium statistical mechanical model of tree graphs which are made up of a linear subgraph (the spine) to which leaves are attached. We prove that the model has two phases, a generic phase where the spine becomes infinitely long in the thermodynamic limit and all vertices have finite order and a condensed phase where the spine is finite with probability one and a single vertex of infinite order appears in the thermodynamic limit. We calculate the spectral dimension of the graphs in both phases and prove the existence of a Gibbs measure. We discuss generalizations of this model and the relationship with models of nongeneric random trees.