Short answers, pulled from the story.
Botanically, a nut is a fruit with a woody pericarp developing from a syncarpous gynoecium, and it is indehiscent, meaning it does not open at maturity to release the seed. True nuts are produced by plant families within the order Fagales, including Fagaceae (beech, chestnut, oak) and Betulaceae (hazel, hornbeam).
Hazelnuts, chestnuts, and acorns have hard shell walls that do not open naturally to release the seed, making them indehiscent. They originate from a compound (syncarpous) ovary and are produced by plant families in the order Fagales, meeting the strict botanical criteria for a true nut.
Walnuts and pecans, which belong to the family Juglandaceae, are difficult to classify and are referred to as drupaceous nuts under some definitions. Their classification is contested because their fruits share characteristics with drupes as well as with true nuts.
Nuts supply a relatively large quantity of calories from unsaturated fats including linoleic acid, linolenic acid, and monounsaturated fats. They are also rich sources of B vitamins, vitamin E, essential amino acids, protein, dietary minerals, and dietary fiber, with low water and carbohydrate content.
Tree nut allergies and peanut allergies are biologically distinct because peanuts are legumes while tree nuts are hard-shelled fruits from a different branch of the plant kingdom. Experts nonetheless recommend that people with either allergy avoid both peanuts and tree nuts, as immune responses can generalize across the two categories.
A nutlet is a small nut; in botany, the term can also describe a pyrena or pyrene, which is a seed covered by a stony layer, such as the kernel of a drupe. The older term for the same structure was nucule, though that word now more commonly refers to the oogonium of stoneworts.