You don't say where you are or what kinds of trees might be available to you, so....I will say red cedar or locust or something similar would do for the posts. Usually for a wattle fence the post were just sharpened and driven in. The "weavers" or horizontal pieces could be of a less-durable species, except perhaps a few closer to the ground. A wattle fence less than 8 ft high would not keep out any deer, and that would not be deer-proof, just a bit discouraging to them. Good luck.