Most people are aware that once you've had foods out of the freezer and allowed them to thaw, you shouldn't put them back in your freezer.
But what about frost hardy plants like leeks? I've just harvested some old onions, leeks and carrots and who knows how many times they've been frozen in the ground, then thawed out.
Is this OK?
A raw root vegetable can sit at room temperature or slightly cooler for weeks and months without bacterial growth that will harm you. The same is not true of meat, cooked vegetables, and other things you're likely to have in your freezer. That's where the "don't refreeze" advice comes from - the food may have started to spoil, and you should therefore cook it and eat it quickly before it gets worse. Entirely different from whether a raw root vegetable happened to freeze or almost freeze.
Some vegetables are so much nicer after a frost that my parents had a rule that if you harvested them earlier (because you couldn't wait to eat them or there hadn't been a frost by a meal that called for them), give them a little time in the freezer. Parsnips, for example, are much sweeter after a frost.
I think the advice to not re-freeze anything has more to do with the structure of the produce being broken down by the ice crystals (the cell walls burst from the pressure or the ice crystals cut through) and you end up with a "mushy" (highly technical term) vegetable. This can also lead to freezer burn (parts of the vegetable dehydrated due to water loss). If the onions, leeks and carrots all look OK to you (mites, molds, fungi, etc), and pass the sniff test and aren't soft, then I see no reason the think they would be bad.
Yes, it's safe. Spoilage in vegetables that have had "too much" frost is obvious -- e.g. spinach leaves will be discolored or mushy. Classic advice is to let parsnips overwinter, so that the cold weather makes the plant convert starches to sugars. This works with carrots too, though if they experience too many freeze/thaw cycles, they'll turn to mush (in which case they'll be obviously unappealing to eat!).
In addition to finding this question when looking into this, I came across an article from the University of Illinois Extension, which had some relevant information:
Probably the concern over eating produce after a freeze goes back to one plant – rhubarb. [...] We eat the rhubarb stalks and should never eat the large leaves any time of year. The leaves are inedible because of oxalic acid and oxalate content which can cause poisoning.
In response to low temperatures, oxalic acid increases in rhubarb stalks as leaf tissues begin to freeze.
This is given alongside advice that frostbitten tomatoes should not be canned due to potential bacterial infiltration (made possible by the cold-damage) plus what the the Montana State University Extension's Lynn Paul says:
Frost bitten tomatoes have less acid in them, which is needed to keep the food safe during long storage after canning.
Otherwise, resources I found seem to agree with the other answers here that frostbitten vegetables in general are safe. I'll update this answer if I find other caveats.
