It would actually be possible to train it to one central leader, you'd have to repeatedly cut away unwanted growth. You could dig out the stump fairly easily if it's less than, say, 4 inches across...jiffy is correct that persistantly cutting off foliage will eventually kill the stump, but if you want a chemical solution to your problem, Ortho and many other companies sell Brush and
weed Killer, Stump Killer, etc. Don't waste your time with the "stump rot" or "stump remover" products that require drilling holes, if you read the directions thouroughly, you'll notice that the ultimate goal is to make the wood rot enough that it will burn...not a good idea that close to the house!