We just moved into a new house. There was a sickly looking tree in the backyard. I cut it down but now there are little branches coming out of the ground in the area the tree use to be. I think it is the old root system trying to stay alive. I really liked the location of the tree, one of the neighbors said it was an apple tree. I cut it down in the winter, had no idea what it was. My question is if I wanted the tree to come back could I let one of these branches keep growing and it would turn into a tree again or is this just some sort of last ditch effort of the tree trying to stay alive but would not turn into a tree again.
There is one possible problem. If it is an apple tree, and if you cut it really low down, close to the ground, the growth you're getting may be off the rootstock itself, rather than the original, grafted apple variety. If the stems you get turn out to be thorny as they mature, then you'll have a wild apple rather than a variety that you might actually want to use. Some apples are grafted much higher up on the stem or trunk, so even if you didn't cut low down, you may find all you've got is a wild apple.
I'd be inclined to poison the stump with a stump killer, together with any major roots you can find, or else bore out the stump to a depth of 18 inches, taking some of the major roots too. And then plant an apple tree of your choice, if that's what you want.
It would grow back into a tree again. By cutting it down in winter, all the energy was stored in the roots. So when spring came, a flush of new growth happened.
What you have done is called Coppicing
Coppicing is a traditional method of woodland management which takes advantage of the fact that many trees make new growth from the stump or roots if cut down. In a coppiced wood, young tree stems are repeatedly cut down to near ground level. In subsequent growth years, many new shoots will emerge, and, after a number of years the coppiced tree, or stool, is ready to be harvested, and the cycle begins again.
Coppicing maintains trees at a juvenile stage, and a regularly coppiced tree will never die of old age—some coppice stools may therefore reach immense ages. The age of a stool may be estimated from its diameter, and some are so large—perhaps as much as 5.4 metres (18 ft) across—that they are thought to have been continuously coppiced for centuries.
Pollarding is similar:
Pollarding is a pruning system in which the upper branches of a tree are removed, promoting a dense head of foliage and branches. It has been common in Europe since medieval times and is practised today in urban areas worldwide, primarily to maintain trees at a predetermined height.
It will probably grow back as a wild apple if it was originally a grafter tree. If it does grow back, instead of poisoning the stump, just graft a new Apple variety of your choice onto one of the sprouts! You will get the choice of your apple variety, and it will grow WAY quicker than a new grafted seedling or sapling, because it ALREADY has a mature root stock, and now it can put all it's energy into above ground growth of the grafted part! And if you don't know how to graft, hire a tree expert (couldn't cost more than $50) or just google it.
I know I'm three years too late, but for the benefit of anyone else who might run across this exchange: Yes, an apple tree will regrow from its roots if cut down. I had this happen with two old apple trees that had to be cut down because they were riddled with carpenter ants. Within five years, I had two 15' trees where the old trees had been and they bloomed and fruited. Because I didn't intend to use the fruit, the root stock wasn't an issue.