In subalpine forests dominated by Abies species in Japan and northeastern United States, trees show traveling wave of regeneration with many striped zones of tree dieback, moving downwind at a constant rate. Previous theoretical studies have demonstrated that a very simple model can generate wave-like spatio-temporal patterns of tree regeneration in a lattice-structured habitat with each site occupied by a cohort of trees. A cohort taller than the average height of its windward neighbor experiences stand-level dieback in the next time step and the height becomes zero. Otherwise the cohort increases its height at a constant rate. Starting from a random initial pattern, this simple deterministic model can generate a saw-toothed pattern that moves downwind at a constant rate, but the distance between adjacent dieback zones has a large variance. In this paper, we study the effects of ''noises'' in tree dieback rules in two forms which help to generate more regular patterns: (1) additional random disturbances at a low rate, which change the size of ''clusters'' (defined as a group of cohorts between adjacent dieback zones) by splitting a large cluster into two or by merging a small one with a neighbor, and (2) the stochastic rule of tree dieback, represented by the probability of dieback in unit time being a sigmoidal function of the difference in the tree height between the site and the windward neighbors. These noises are effective both for one-dimensional and two-dimensional models, but spatial patterns are much more regular in the two-dimensional model than in the one-dimensional model.