Answer: Queen Anne's lace is a member of the carrot family and as such as a deep taproot. so, even if you kill off the tops with vinegar or boiling water, the root will have enough stored energy to resprout new tops. Digging is best, but second best is cutting off the tops and then covering over the soil with cardboard or several layers of newspaper to exclude sunshine. Without sun, no photosynthesis can take place and the roots will resprout tops but they will die out and will not be able to restore the energy to the roots. Eventually the plants will die out.
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