We sympathize. We had a large sycamore which invaded the vegetable beds with its roots. While we avoid tilling in most cases, once a year we do fork over the areas that the roots get into, and pull out what we can. You could sink something into the soil—wood, stone slabs, sheet metal, something like that—around the vegetable beds where the tree roots try to come it to divert them. Failing that, pay special attention to feeding and watering vegetables growing on top of the roots to try to minimize problems with competition for resources. A good thick layer of compost or well-rotted manure on top of the soil every year helps, too. This does raise the question of whether you’re feeding and watering your vegetables, or the tree, but with good soil there should hopefully be enough to go round.
We sympathize. We had a large sycamore which invaded the vegetable beds with its roots. While we avoid tilling in most cases, once a year we do fork over the areas that the roots get into, and pull out what we can. You could sink something into the soil—wood, stone slabs, sheet metal, something like that—around the vegetable beds where the tree roots try to come it to divert them. Failing that, pay special attention to feeding and watering vegetables growing on top of the roots to try to minimize problems with competition for resources. A good thick layer of compost or well-rotted manure on top of the soil every year helps, too. This does raise the question of whether you’re feeding and watering your vegetables, or the tree, but with good soil there should hopefully be enough to go round.