The Adobe watermark can be removed from a Premiere Elements tryout file after you purchase the product with its serial number. Then you open the purchased product, open your tryout project (project.prel) file, go to Timeline Menu/Delete Rendered Files. This it it.
(In my case, I found that the Adobe Watermark disappeared from the tryout project.prel by just opening the tryout project in the purchased software.)
Keep in mind that you cannot remove the Adobe Watermark from anything that you have exported trom the tryout (before or after purchase of the serial number).
Please let us know if you need clarification on anything written.
Full instructions are also posted in a link on the second page of the FAQ
The Tutorial Links Page http://forums.adobe.com/thread/1275830 may also help
Plus the Premiere Elements TIPS List (6 pages of Tips as of September 2013)