Hello,

You do have to remember that most of the tree's roots are within the top 12 inches of soil, so wherever you dig under the tree you will most certainly be hitting roots.  These roots can extend well beyond the drip line, reaching about 1 1/2 times the height of the tree.  This said, gardeners do plant under trees.  You do have to take care not to change the grade around the tree's root system and do not add more than a few inches of soil over the tree roots.  Then, careful digging can help avoid damaging large roots.  Choose smaller plants to plant so that planting holes can be smaller and go between the larger roots.  And know that your hackberry tree will need additional water this summer to help make up for the lost roots. 

For more information:

https://lancaster.unl.edu/hort/articles/2011/PlantingUnderTree.shtml

I hope this is helpful.  Thank you for contacting Ask Extension.

Ruth