We solve the growing asymmetric Ising model [J. Sienkiewicz, K. Suchecki, and J. A. Hołyst, Phys. Rev. E 89, 012105 (2014)] in the topologies of deterministic and stochastic (random) scale-free trees predicting its nonmonotonous 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 (lnlnN)(-1), where N is the number of nodes in the tree.