We study the statistics of branching polymers with excluded-volume interactions, by modeling them as single self-avoiding trees on a generic regular periodic lattice with coordination number q. Each lattice site can be occupied at most by one tree node, and the fraction of occupied sites can vary from dilute to dense conditions. By adopting the statistics of rooted-directed trees as a proxy for that of undirected trees without internal loops and by an exact mapping of the model into a field theory, we compute the entropy and the mean number of branch nodes within a mean-field approximation and in the thermodynamic limit. In particular, we find that the mean number of branch nodes is independent of both the lattice details and the lattice occupation, depending only on the associated chemical potential. Monte Carlo simulations in d = 2, 3, 4 provide evidence of the remarkable accuracy of the mean-field theory, more accurate for higher dimensions.
Marcato et al. (2025) studied this question.