Brief history. My brother gave me a sack full of lemons from a lemon tree he has at this house. I planted 6 seeds from one of these lemons 2 years ago. One survived. This winter, I left same outside in a planter box that is about 18" tall. Froze few times here where I live a few times this past winter. No leaves on it whatsoever. A stick. I kept it damp anyway. Now I got shoots with leaves from apparently rhizomes in the rootage.
Should I cut down the "stick" or wait awhile?
If you got hard freezes, the stick is probably dead. Citrus is good down to about 28 degrees, as long as it does not stay down there for too long. Below that and it's toast (or more like "popsicle").
Of course there is no harm in waiting. But I would guess everything that was above-ground got killed. You were fortunate to not have a deep enough freeze to freeze the entire pot, so the roots lived. (Your citrus is now more like a perennial plant, coming up from the roots each year.)
If you want to keep this thing alive long-term, it would be best to find a spot for it to put it when a freeze is expected where it will not be subjected to freezing temperatures. If you are in a warmish winter area, an unheated garage or shed could be enough. If it's got a window, you can actually put it in there all winter. Citrus are evergreen so they don't want to go completely dormant in the winter. They will live on very little light, though I think they need to have some.
If you are in a hard winter area, you'll probably have to bring it inside the house. Again, try to keep it by a window that gets as much sun as possible. I keep our orange tree in a south-facing window all winter. It tends to think spring has arrived due to the sudden warm-up when I move it into the house around October, and it starts blooming. (I manually pollinate and we do get a few oranges from this.) Then when it goes outside again the next spring, it thinks it's spring again (this time correctly) and blooms again.
One caveat is that after it's been indoors a while, it will not be ready for full sun immediately. After a lengthy indoor stay you are supposed to put it out for short periods then longer and longer each day, so it can become accustomed to the more intense sun and avoid having its leaves scorched.
Me, I'm too lazy for that kind of routine (especially because this tree and pot weigh something like 50 pounds!), so every spring I just put it out there once and for all to sink or swim, and its poor leaves do get scorched. I try to put it out when we are expected to have a lot of clouds for a few days and the outdoor temp is not too high, and cross my fingers that this gives it a little extra adjustment time. But citrus is amazingly resilient (if it's kept above freezing :-), and it has always bounced back from its spring sunburn. (The scorched leaves of course never "heal," but it puts out a lot of new leaves.)
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