I'd agree with Fleecewife about 'sun thirsty' things like tomatoes etc. not being happy (unless you particularly like green tomato chutney!), but I'd say give things a try; you've got nothing to loose by trying and it will be informative. Our old neighbour had a patch that was completely shaded by huge oaks (literally right next to the patch) on one side, the house on the other and massive rhododendrons on the 'sunny side', however, her veg used to completely out perform our sunny, more open side despite what you would think by a visual comparison of the two sites. She was in her 80s so it's not as if she worked it as much as we did either! Consider the aspect - what side is the tree on, where does the wind come from, does it get any sun at any point? You might find that it has benefits you hadn't thought of.
We have a sycamore overhanging our new veggie patch (on the east-ish side), but I find it gives the plot that bit extra shade that actually means the veg is far happier for it than it would have been otherwise what with this hot spell we've been having (it doesn't get the sun till around the middle of the day). I've got cabbages, beans (runner, French and broad), celery and lettuce on that part of the plot and they are thriving.