Thanks to all for the responses. I took Chuck's advice and went to gardenweb, which is a very good site. There was an existing thread which exactly answered my question. For the benefit of others, here is the answer.
"If you know a little about grafting you would know that the trunk is a vertical rootstock which gives your weeping plant height before it starts weeping. In other words, it keeps it from looking like a haystack. The part that weeps is a cutting that is grafted high on the top of the rootstock. There should be a distinct difference in the bark and there should be a scar where the two compatible plants joined and healed together. If your vertical sprout is a part of the graft it won't grow far before it begins to weep again on its own. However, if the vertical sprout emanatates from below the graft then it will continue growing vertically and will become bigger and bigger in girth and the weeping part will live a few more years but stop growing."