I always use fallen branches, as I don't believe a healthy tree should give up its branches for my little project. I scour my yard and the woods, and I'm not picky about the type of wood I use. I also keep an eye out for neighbors who might be pruning their trees; they are often happy to be rid of branches. If you do use this green wood, I suggest letting it dry for a few months to minimize shrinkage or twisting, which could pull apart the glue.
Try to gather many more branches than you think you will need; having a number of choices helps. To give the branches a smooth look, strip the bark and sand the branches before you start putting them together.
When I build one of these tuteurs, I try to keep in mind that the structure is supposed to be rustic, so flaws are OK -- anyway, they will all be covered by vines. As you might imagine, the project can be built to whatever Dimensions your plants require or your branches allow.