Try lettuce. May be you shoud prune some branches off the trees so at least you can have filtered sun light. Spinach definetly need the sun. My neighbor grows tomatoes in the shade .They are not the gratest but they do get fruits.
I have Asparagus in the shade.
If you are in a Nothern USA you can grow more vegetables since early season there are no leaves in the trees.

I have once grown asparagus in the shade and to my surprise it came out very well. We had it for dinner and the asparagus was delicious. I can't imagine that tomatoes grow too well in the shade but maybe you can give it a try.

Rhubarb will grow in part shade. It is a perennial plant and only the stems are eaten. The leaves have a high concentration of oxylate which makes them poisonous.

Radishes (the little Western ones) only take a few weeks to form up. and can be planted very early, before the trees' leaves develop. Peas can also be planted before the last spring frost.

There is a climbing squash called "climbing zucchini" which will climb tall shrubs and have a long fruit hanging down. It can be eaten like zucchini in summertime or like winter squash when fully ripened.

Gooseberries will grow in part shade, as will raspberries.

You could also try putting pots of vegetables in front of your home, in the sun.

Mint, lemon balm, and many other mints that can be used for tea will grow in part shade...check over at the herb forum for more information.

The people in the vegetable forum may be able to give you more ideas as well.

Sheva,

There is a source book titled Cornucopia, published by Kampong Publications. It is filled with food plant information, products and sources. I use it as a springboard to do web searches on this sort of thing.
J.L. Hudson, Redwood City, CA sells it, among others. He has a web page.

Mike519