The mulberry tree
Where I live in Le Marche most old farms have one or two mulberry trees growing in front of the house. These not only served as shade in the summer but more importantly were the habitat of the silkworms, which were an important part of peasant economy as recently as thirty years ago.
Mulberry trees grow fast, and provide the most excellent dense shade. They are available without berries, with white berries, or with black berries. 
These mulberry trees, which form a walk at the Abbadia di Fiastra near our house, are a mixture of all different types, so that in places the ground is covered in squashed purple berries, in places clean, and in places with the white berries. If you are planting the tree over a wooden deck or paving I would recommend going for a tree with no berries.