We have a similar issue with trees adjacent the veg plot.

There is a row of mature silver birch trees next to (North of) the veg plot.
I've hit shallow tree roots while rotovating the plot within 15 or so feet
of the trees.
Most of these roots (up to 1" diameter) I removed last year and the trees
don't seem to be suffering.
I discovered that vegetables growing within around 10 feet of the trees did
badly, the ground was too dry for them without excessive watering, so there
must be more tree roots running below them deeper than rotovator depth
sucking the soil dry.
Veg 15 feet or more away are unaffected by the trees.
In conclusion - I'm going to grass over the ten feet or so between the trees
and the veg plot. Can also use the area for compost bins, garden shed etc.


--
David
.... Email address on website http://www.avisoft.co.uk