Are these food items in the same category as peanuts, walnuts, etc? If I tell a patient to avoid all tree nuts, does it mean coconut, pine nuts, nutmeg?
Differences: seed has haploid germinal reproductive cells and secured inside fruit; Nut is any dry part of fruit you can eat such as walnut (seeds plus edible embyro)
This kind of botanical analysis - seeds vs nuts - is not relevant to allergy. Neither is the 'tree nuts' label - it is traditional, but not at all useful. What usually matters in allergic cross-reactions are the relatedness of plants (ie do they belong to the same plant family), not whether they come from trees or bushes or whatever, or what botanical structure they are derived from. However, for reasons unknown, there is a great deal of cross-reaction between the various items we call (casually) 'nuts', regardless of relatedness. Pecans and walnuts are related, as you might guess from their looks, but no other nuts are related to one another. Brazils, almonds, walnuts, cashews and peanuts and so on are all from entirely different families - so they shouldn't cross-react. But they often do. So people with peanut allergy may also react to walnuts or almonds or other nuts. But note that not all patients do, and sometimes people are allergic to just one type of nut, or to some and not others. So, as far as your first question goes, one can only work from the clinical evidence - which is that very few people with nut allergy ever react to pine nuts, coconut or nutmeg. The patient should cautiously try a very small amount to check they are OK.