Meh, trunk vs branch is nothing, and really after all there is no serious difference between the two... what you SHOULD be impressed with is that the meristem in the leetle dormant bud points on that twig can (depending on the circumstances) develop into twig/branch OR leaf OR root OR even flowers
It is controlled by plant hormones (auxins, giberellins, cytokinins, stuff like that) which in turn are controlled by various factors such as light/dark, gravity, and what other nearby tissues in the plant are doin'.
It has been a bit too long since I taught intro bio type courses for me to feel confident doing a summary

Here is Wikipedia's entry on plant meristems
http://en.wikipedia.org/wiki/Meristem
Plants that tend to grow with a single main axis of growth -- pine trees, for instance, as opposed to messier shrubbier things with several-to-many main trunks -- do that because of a thing called apical dominance. Whichever bud is endmost on a plant sends out hormonal signals that somewhat suppress the growth of nearby buds/shoots, so that the topmost shoot grows preferentially and you get a traditional single-trunked tree shape. (In species that DO that, of course -- plenty don't). Although, if you notice in a lot of plant species the branching pattern is basically fractal -- the branching pattern within subshoots of a particular branch mimics the overall architecture of the whole plant -- which also contributes to the ease of "a twig becomes a trunk".
There have GOT to be some good 'plant growth form and development regulation 101' type explanations on the web, but a whopping three minutes spent with google failed to find any, perhaps someone else can make suggestions?
Good luck, have fun,
Pat