It depends on how your rose bush was produced. Most varieties of roses are propagated either through grafting or cuttings. In grafting, a twig from a plant of the desired variety is grafted onto an existing rootstock, while cuttings are placed in soil or growth media to take root directly.

Grafted plants will have a knob or bulge on the trunk where the graft was attached. If the new growth is coming from above this point, it will produce Ruby Anniversary roses. If the shoots are coming from below the graft, they will reflect the genetics of the rootstock, and the roses will be completely different. In the image below, the graft was attached where the main stem changes from woody to green:

http://andysworld.org.uk/blog/wp-content/uploads/2009/06/rose-sucker2.jpg

Cuttings, on the other hand, are genetically uniform, and any growth from the original plant will produce the original type of rose.

If the new growth is from a cutting or from above the graft, encouraging a single main stem will eventually regrow the plant's original treelike appearance. If the growth is from the rootstock, then all bets are off; there is no way to tell what the flowers will look like or how the plant will grow (tree, shrub, climbing, etc.). You might end up with something attractive, or you might need a new rose bush.
Source(s):
http://www.backyardgardener.com/plantname/pda_1453.html
http://www.rooting-hormones.com/rose.htm

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