Like most people, I grew up hearing that you didn't want to plant a garden too near a black walnut tree or certain oaks because there was "something" in the trees that killed the plants beneath it. When I got older and did some research, I learned about allelopathy and knew that the "something" was one or more chemicals produced by certain parts of the tree. I also added a few more species to my "no-no" list -- like eastern red cedar, bay trees, and certain maples; and found that the part that effects other plant's growth can vary from species to species -- cedar trees, for example, produce their toxins in the leaves, so other parts are fine around other plants. (So, you CAN use cedar boards and logs for hugelkulture and raised beds.) What I didn't know until this morning is that some plants actually do better under certain allelopathic trees!
I just read a couple of studies (one from the U. of FLorida and another study) where they grew several known allelopaths -- including, believe it or not, broccoli -- in conjunction with crops like peas, wheat, rice, rye and corn. They found that (just a couple examples) under walnut trees, corn production went down, but under Leucaena, wheat and tumeric went down but maize and rice actually increased.
http://edis.ifas.ufl.edu/hs186
The second study sprouted and grew peas and corn watered with solutions made from cedar foliage. Apparently peas are not affected by cedars, but corn -- though it will sprout, does not achieve as great a biomass.
http://escholarshare.drake.edu/bitstream/handle/2092/956/Poster%2020.pdf?sequence=1
There is a lot more on this out there and I am going to read it avidly. I have acres of red cedar and lots of walnuts, oaks and other problematic trees. This may have huge impact on my future garden!