The fields of course should be left to the farmers (corporations) to plant trees between their fields, because after all it is their property, after all, however I am sure they will demand some kind of compensation for growing additional trees that will benefit the most from.
We have extreme problems with current farming techniques. Major corporate farming techniques are derived from the perspective that greed is good and the commons must be exploited at all costs because all resources are limited. The reality is nature is all forgiving and when we farm with the objective there is plenty for all as in this wonderful TED talk, everyone benefits, not only the environment, other animals and people as well.
We must move towards the perspective there is plenty.
If we started planting drought tolerant plants, such as the above date palm, or the following plant that likes salt water or non-edible plants like Nerium which are extremely easy to grow from cuttings but extremely toxic or Salvation Jane which is also toxic but edible by live stock in a pinch and makes good ground cover. Jasmine is also quite good ground cover but not so easy to strike from a cutting but is edible. Geraniums are super easy to strike (just break a branch off and stick a few of the nodes in the ground), edible as well and extremely drought tolerant.
But ideally we want to grow plants to heal the earth and ourselves. Our selfishness and greed is hurting ourselves mainly.
I believe we have to be taught greed but then someone said haven't I seen the feeding frenzies of wild animals? Animals know scarcity too but they are not in the same position we are in where we can learn that if there is plenty for all we can turn our damaged planet back into a paradise.