Often the connection between the new growth and the stump is weak. Also, the stump will rot, and although the new growth will compartmentalize against this, the scenerio is not good. To answer your question, yes, you can let ALL suckers grow for a season and pick the strongest. However, a sucker coming from a dead stump is hardly the best idea for a new tree.
Hortster

I see questions about stump sprouts fairly often in this forum--it is an important and interesting issue. I have had a lot of experience with trees from stump sprout origin, and have read a number of silvicultural articles about it. Here are some important points.

First, different species of trees have different potentials for successful stump sprout regeneration. With some species, such as black cherry, stump sprouts are almost universally successful, and often result in magnificent trees, strong, vigorous, and basically as good as trees of seedling origin. Other kinds of trees are a bit more "iffy," such as red maple. But even red maple stump sprouts can result in good trees.

I really don't have any information about birch trees, and it may make a difference which kind of birch you have.

But here are some general guidelines. First, for the first year, leave all the sprouts to grow. It looks like you are in the second year now, so here is what you should do. Remove all the sprouts that are near the top of the stump--those near where the point where the tree was cut off. Remove these even if some of them seem to be the most vigorous. Next, look for those that are lowest on the stump. Any that are growing at or just above the root collar, if you can identify that, are those that have the best potential. If any of these are among the most vigorous, and come out from the stump fairly straight, these are your best potential trees. If you can find anywhere from two to as many as five of these, leave them all for now. But, if any two or more of these "best sprouts" are growing close to each other--within 3 inches or so--remove all but one from each group/pair. Now remove all the other sprouts.

Next year you can remove all but the best two or three. Then you can decide if you want a clump of more than one tree or just one. If you want just one, wait one more year, and then remove all but the one that seems to be the strongest. If one is lower on the stump--closer to the root collar--that should get some priority, unless some other one that is nearly as low seems clearly more vigorous.

If you do these things--and have sprouts at or near the root collar--your chances of getting a really good new tree this way are very high. Stump sprout trees do not necessarily have the rot from the stump migrate into the new tree--ever--especially if they grow from a point very near the root collar. And, although they can easily be broken off when they are very young, later these trees can be very strong and are not more likely to break than other trees.

Good luck!

--Spruce

I generally agree with spruce and find their advice excellent, but not all trees stump sprout.

Birch are a ruderal spp and stump sprout to the extent that they will do so successfully when conditions allow and the stump sprouting has a chance to ensure fruiting. This fact is key about your tree: birch are not a long-lived spp. If you wish to expend time, energy and money to try and ensure a short-lived tree resprouts for 15-20 years, that is intersesting but not logical. Replace with a long-lived seral tree.

Dan

It is possible. I would take a bunch dig them and pot them up you are likely to get some that survive. You also can prune parts of them and put them in moist sand with rooting hormone and root them. Also just digging up a bunch of roots and breaking them into peaces and putting them in dirt will work. I have grown Aspen trees from roots before.

Aspen is a whole different tree entirely. Birch trees don't start from root cuttings. It's all a person can do to contain an Aspen.
Spruce's advice is right on, as is Dan's and the Hortster's.
Mike

Birches can look good as multi-stemmed clumps. I think it would look better selecting a few stems rather than just one. It might not last many years but could be attractive for some years. However, since the post is 10 years old I don't think the OP is likely to be reading this.

Nonexistence of OP in this discussion notwithstanding, I mildly disagree with the notion that birch are always and everywhere a short-lived tree. Surely that is generally true, but there is at least one major exception worth noting, that of paper birch and/or yellow birch growing in close proximity to the Great Lakes. These large bodies of water-known locally as nature's air conditioner-so modify the air that some trees can live for considerably longer than the norm. Likewise, in deep, cold swamps in the north, birch occasionally go on for quite some time, gaining surprising size in the process. So, that pot-bound "clump birch" from the garden store, rudely dumped out in a suburban Chicago lawn somewhere.....yeah,, that thing is going to struggle to make it past twenty years or so. Put that same species of tree, albeit one from seed, in a white cedar swamp like up at my land......you will be amazed both at how long these things can live an how large they can become.

+om

well go figure tom.. they grow better in a native environment ... without the instant gratification of some homeowner ...

ken