Nuts can be an excellent source of plant-based nutrition, of course. When choosing trees or shrubs for your garden, it is worthwhile thinking about nuts as well as about fruits. The options available to you will of course depend on where you live. In my climate zone and on my property , there is a short list of nut trees that might be worthwhile.
Top of the list is most definitely hazel (Corylus avellana). This is one promising permaculture plant for a number of locales. Hazelnuts are an excellent protein and oil crop, and are also great for attracting wildlife, making hedgerows, and providing wood for many projects. Hazel works well in coppice systems.
There are several hazel trees in the area around our property (though the nuts are small and to be honest, are eaten by wildlife before we get a look in). I am planning to add some hazel at some point for edible nuts, and am considering looking to hybrids with Corylus maxima which have larger cob nuts. In the US, Corylus americana is the native hazel species, with similar benefits.
Pine nuts are another avenue I have been looking into. Pines in my area do have edible seed. But these are generally small, and again, there is a lot of competition from wildlife. Pinus siberica, Pinus cembra, Pinus edulis, and Pinus koraiensis are just some of the options that can be considered. (The last of these is considered to produce the most worthwhile nuts in colder climate zones.)
Beech nuts are foraged here, but are small and difficult to harvest. Again, wildlife often gets there first.
Walnuts are grown further south, but don’t do well here. But if you live in the US, or in warmer temperate regions, these are on of the most promising options if you want to grow nuts. Black walnuts, white walnuts/ butternuts, heartseed walnuts, buartnuts (hybrid), Manchurian walnuts, California walnuts might all be interesting options, depending on where you live.
Sweet chestnuts don’t tend to ripen well here. But again, in other areas, there are several chestnut options to consider: sweet chestnuts in Europe, and American chestnuts (or Chinese chestnuts, or hybrids that are blight resistant) in North America. In zones 4-7, Chinquapin is another related nut to consider.
Hickory (or Pecans further south) are another interesting avenue to explore in the US. And further south, almonds and pistachios are of course other options. Only in far southern reaches can cashews and other warm climate, subtropical or tropical nut varieties by grown.
If you are looking to try something a little different, bladder nuts for both Europe and the US could be options. And the non-native Yellowhorn could be an interesting option where sweet chestnuts cannot be grown (as the nuts of this shrub or small tree are said to have a similar taste).
Of course this is my no means an exhaustive list. But it could help you begin to consider the nuts that might be grown in your garden.