The fruit of the beech tree, called "Beechnuts", are found in the small burrs that drop from the beech tree in Autumn. They are small and triangular, are edible, have a sweet taste and are highly nutritious. (~ 20% protein and also ~ 20% oil content). However, they do contain organic substances which are slightly toxic (it has been reported that eating approx. 50 nuts may make you ill) so that they should not be eaten in larger quantities.
I don't think that I have ever seen them offered for sale, though I do recall beechnut chewing gum as a child!