  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.
