There are some field-level techniques that you can do, which I learned about during my research.
First is to have barrier and non-barrier plots. The barrier is made by digging (preferably mechanically) a trench along both sides of the plot (say, 1 m belowground), and lining it with black plastic sheeting up to the surface. This helps to prevent interspecies root competition.
Second, in order to measure for water differences, you can use a couple of techniques. For water, you can install ceramic-tipped lysimeters, which draw up the water, which can be tested. (You can make the lysimeters yourself, or buy them commercially). You can also install tensiometers, which will show you how dry the soil is in various places and depths.
Third, to measure for nutrient differences, you can use N15 isotopic-enriched fertilizer. This can be bought commercially (it's pretty costly). You apply the N15 fertilizer to different plots, and you can follow the fate of the fertilizer into soil, water, crops and trees, by having those samples analyzed in the lab (the N15 will still be present in the samples, even months later). There may be some other fertilizer method that could yield similar results.
Anyway, there are other techniques I'm sure, but those worked pretty well for our research. You can read more about it in some of our research:
Water competition:
Article Competition for water in a pecan (Carya illinoensis K. Koch)...
Nutrient competition (and evidence for positive phytoremediation effect):
Article Competition for 15 N-labeled fertilizer in a pecan (Carya il...
Hope this is helpful. Wish you the best with your research.
I agree with Samuel you can use several technique for study below ground interactions. I found useful measuring root distribution and water uptake of the component using neutron moisture meters or others devices. However It alwasy depend on what you really want to study, hypothesizes and your resources available. I am attaching a paper of my research on below-ground interaction between leucaena and grass.
You must analyzed the effect of the species on the soil properties, the physiology and yield. So, soil samples will be taken at around isolated individuals of species following the East-west and north-south direction and at two levels of distance: under and out crown. Submit the samples to a chemical analyse to determine pH, Organic Carbon, Nitrogen and Organic matter content and the ration between Carbon and nitrogen (C/N). Concerning the productivity, settle the squares of yield at around individuals of species: squares of yield in each level (under and out crown). In each square, the height of stalks, the number of suckers, the biomass as well as the yield were estimated.
You raised an exciting point. If you want to study the below ground tree - crop interactions, there are several parameters you can consider. You can study the changes in competition for water and soil-nutrient, root distribution, changes of soil biotic factors compared to mono-culture system.