I live in the southeast and have managed commercial landscape operations from texas to fla. as well as made I don't know how many personal veggie and flower plots. very seldom have i not found your situation, when preping a new spot.
unless u have bushes next to ur plot, very unlikely to hurt them and it is extremely unlikely to hurt a tree and it is very likely it is either tree roots or tree like shrubs like crepe myrtle. small roots like you are experiencing can run forever, it seems, certainly well past the water line of trees.
the best method is to just chop em up. you don't want to just process actualy planting spots, u need to always prepare the entire area, so if it is with a pair of snips or an axe or machete or a roto tiller, just chop em up, remove the pieces and prepare as you would anywhere else.
I plant in deep, double dug beds, anyway, incorporating organic matter such as compost, rotted manure, peat ... and then bed it up. very seldom is the encroachment of the roots more than an occasional problem, if bed prep is done well and a foot deep. be careful w/ too much rototilling, it creats a shallow hardpan, a hardened subsurface "floor" only 6 or so inches below the surface, making it hard for your plants roots to penetrate.
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