I would go for fast growing early spring green vegetables for salads, such as lettuce, spring onions, anything that is ready to eat within the sunny three months, if it is too cold then put them under polythene tunnels or glass.
I would also consider runner beans and start them early as possible indoors and plant them out as soon as possible, as they will have a relatively short life and will start to fail with the sunlight and dry. The other issue with being under trees is the ground will likely be poor and the trees will grab the available moisture. Invest in 'grow-bags' or make your own with long strips of polythene about 2 metres wide, dig out V shaped trenches and line them with the poythene - then fill the trench with a growing medium that is the best you can afford in bulk. Planting in these trenches keeps your plants and the trees apart and the polythene should last a few years under the ground. If you are really smart you run each trench slightly downhill and then you can water them easy from the top and letting the water migrate to the bottom. Sitting by a hose for 30 minutes with a cold beer is preferable to wandering around with the hose on a hot day !