There isn't a setting for this, but I think it is tied to the file. The splitter bar is what makes the tree two. Try opening a new file, then move the splitter bar up to the top, then save the template out. That should save it for every new part. This wouldn't affect the older files, however. I hope this helps.
There is an option in General or Tree Settings the former I think to autosize Property Manager when there is a split. Haven't used SolidWorks to recently but think you can double click to maximize one of them.