Answer: There's no easy answer to the problem, including removing the trees. I think you would miss them! Tree roots are naturally attracted to rich, moist soil so you can't blame them for invading your garden area. You can 1) sink a barrier into the ground on the tree side, 2) plant in a raised bed rather than at ground level, 3) line the sides and the bottom of your garden with weedbarrier and/or screening material.
Option one can be accomplished by digging a narrow trench and laying fiberglass or corrigated panels on their sides. These are typically used as patio or carport coverings and come in 24-30" widths and 6-12' long. They make great barriers for containing bamboo roots and for keeping out tree roots.
Option number two is to simply build a structure and raise it above ground level. Option three can be used for a ground level or a raised bed garden.
A final thought - can you move your garden to another part of the yard??
Good luck with your garden, and with your trees!
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