Ed said:

It's my understanding that if you graft a branch up side down, it will assume a weeping structure. The physics of it escapes me.

Nope.

In a normal plant the shoot tips will grow up, and the roots will grow
down. It's a process called geotropism. There are subtle variations:
secondary branches and secondary roots don't grow straight up or
straight down; they grow at angles. And rhizomes grow horizontally.
Amazingly, scientists don't fully understand how plants do this. It
has to do with hormone concentrations within individual cells, and with
starch granules that sink with gravity and allow plants to tell "up"
from "down". At any rate, weeping cultivars are mutants. The shoots
don't respond in the normal way to gravity. If you graft the branch
upside down, it doesn't matter; up is still up, and down is still down,
and the branch will respond however its genes tell it to.

Nina

Mark said:
From your explanation, I assume that "weeping" trees have unique
hormone levels that cause branches to droop rather than grow upright.
So ... are weeping trees actually a third form of growth (apically
dominant, basally dominant, and weeping) ?

There are several mutations that cause "weeping"; and they seem to have
more to do with the plants's response to hormones rather than to
hormone levels themselves. I don't know what you mean by "basally
dominant"; all plants are apically dominant to different degrees.

And ... I ran across another "SuperThrive" like product called
"MegaGro" which claims to contain a substance called Potassium Gibberelate.

Gibberellin is a real plant hormone which has many various effects on
plants. There's a fungus that causes a disease of rice called "foolish
seedling disease" (probably sounds better in Japanese), making
seedlings grow tall and spindly. The fungus produces an artificial
gibberellin, changing the growth of the plant. Gibberellin is
sometimes sprayed on seedless grapes to make them bigger; on radish to
retard bolting; on fruit trees to increase fruit retention, etc. etc.
etc. It has its uses, but it's not miraculous. There's a product
called "Bonzi" (seriously) that makes ornamental plants more compact.

Nina