You may want to do some photoshop work on the image before you give it to Illustrator live trace. Simply posterize the color of the image to create a nice simple image for illustrator to outline.
Using Photoshop to create selections and convert them to outlines can often produce better results than the trace tools, since you can adjust/modify the selection with a number of options. I usually get many fewer useless anchor points that way. Outlines can be copied & pasted into AI for final cleanup (better vector tools).