  We revisit the random tree model with nearest-neighbour interaction as
described in previous work, enhancing growth. When the underlying free
Bienaym\'e-Galton-Watson (BGW) model is sub-critical, we show that the
(non-Markov) model with interaction exhibits a phase transition between sub-
and super-critical regimes. In the critical regime, using tools from dynamical
systems, we show that the partition function of the model approaches a limit at
rate $n^{-1}$ in the generation number $n$. In the critical regime with almost
sure extinction, we also prove that the mean number of external nodes in the
tree at generation $n$ decays like $n^{-2}$. Finally, we give a spin
representation of the random tree, opening the way to tools from the theory of
Gibbs states, including FKG inequalities. We extend the construction in
previous work when the law of the branching mechanism of the free BGW process
has unbounded support.
