Growing spin model in deterministic and stochastic trees

We solve the growing asymmetric Ising model [Phys. Rev. E 89, 012105 (2014)] in the topologies of deterministic and stochastic (random) scale-free trees predicting its non-monotonous behavior for external fields smaller than the coupling constant $J$. In both cases we indicate that the crossover temperature corresponding to maximal magnetization decays approximately as $(\ln \ln N)^{-1}$, where $N$ is the number of nodes in the tree.