With surfaces, we can only remove a vertex with a zero count if its parent triangle doesn't have any nonzero children. We don't really even remove vertices, instead, we collapse triangles into larger triangles.
In this example, within the large triangle we have two vertices with nonzero reference counts. We first reduce the reference count of the vertex on the right. We notice that all its siblings also have zero counts, and none of them has children, so we collapse the four triangles into their parent.
When the reference count of the remaining vertex goes to zero, we begin a cascade of triangle collapses. Now all the triangles and vertices within the big triangle are removed in a bottom-up order.