A Kripke model \(K\) for Heyting Arithmetic (HA) is locally PA, if at every node of \(K\) the attached structure is a model of PA. It is an open problem whether every Kripke model of HA is locally PA. The problem has been solved affirmatively for Kripke models with various special tree frames. Ardeshir and Hesaam give a simpler proof of a theorem that generalizes some earlier results of D. van Dalen, H. Mulder, E. C. W. Krabbe and A. Visser [“Finite Kripke models of HA are locally PA”, Notre Dame J. Formal Logic 27, 528-532 (1986; Zbl 0632.03048)] and K. F. Wehmeier [“Classical and intuitionistic models of arithmetic”, Notre Dame J. Formal Logic 37, 452-461 (1996; Zbl 0871.03027)]. The problem is solved for trees constructed by attaching to every node of \(\omega\) a finite tree from a family of trees with uniformly bounded supremum of lengths of all the branches, and for infinite rooted trees whose any set of pairwise incomparable nodes is finite.