I found this paper interesting in terms of hypotheses and adopted approach but I have some concerns related to two points:

1) definition of established species. The authors focused on established species but did not provide a clear definition on how these species are identified. In particular, I do not see a clear separation between the concept of alien species and established species. I guess that the second are a subset of the first, but a formal definition should be provided possibly based on some identification criteria. Very likely the final publication of the reference paper can help in solving this issue, but at present is not easy. I have checked the data provided in the anonymised online resource but definition are not clear. In particular, the authors refer to data extracted from the Global Naturalized e Alien Flora database (GloNAF) but again the difference between “naturalised” and “established” is not clear to me.

2) numbers of established species. The entire paper is based on a data set that has been assembled, according to some criteria, but cannot be verified in this moment. This data set is used for performing all the analyses. The figures provided for some islands (e.g. Hawaii or Canaries, of which I know the flora) seem to be quite low with respect to most recent publications. Very likely this is due to the point raised in previous critical point, but I would expect some support on the completeness of these lists of “established species”.

Some minor points:
3) Islands vs archipelagos. Most of the arguments used in this paper refer to islands. However, the data set and the analyses all focus on entire archipelagos and not on islands. In some case, the two can almost exactly correspond (e.g. Taiwan) but in some cases the structure and the number of islands within the archipelago can be a dominant factor that has not properly considered (e.g. Maldives or even Hawaii). To address this issue it is need to focus on island’s data or state that the paper deals with entrie archipelagos, not with islands.

4) habitat heterogeneity. This variable is quantified by using only island area and maximum elevation. This is quite simplistic approach and I think that more sophisticated information could be used with a quite easily access (e.g. land cover).