My daughter was just about 2 when we went in to test her for peanuts. She tested very, very positive for them (doctor said as high as it gets) and he recommended we avoid all other nuts too. He said that the risk of cross-contamination was very high because they're often processed on the same lines and children who are allergic to one type of nuts are more likely to be allergic to other types (though I don't get this because peanuts are technically legumes).
I have a friend though that has a daughter who is allergic to like 10 things, including I think 2 types of nuts. I think they avoided nuts in general for a long time, but now they're introducing other nuts that she hasn't shown an allergy to. I don't know how this works, but she's one of those people who has obviously done her research.
Personally, at least until I figure out this whole allergy and possibly if she passes some sort of age mark that she's less likely to develop further nut allergies, I'll be avoiding all nuts. Nut allergies are one of the scariest because they are one of the types that children can react anaphylacticly or by closing of airways. Maybe I'll think differently down the road, but for now she's so small and I would hate to use her epi pen.