Magneto optical imaging of Nb rings, open rings, and strips reveals that the topology affects the dendritic flux avalanches in the samples. In particular, dendrites crossing the entire width of the sample appear only in the rings. Such dendrites appear when the difference between the applied field and the average field inside the central hole reaches a certain threshold level ΔHth. With increasing applied field, this condition is reached quasiperiodically as a crossing dendrite creates momentarily a hot channel through which flux flows into the central hole, balancing the field inside the hole with that outside the ring. The threshold ΔHth differs in magnitude from the onset field Hth of magnetic instability, and exhibits qualitatively different dependence on the rim width.