There may be a reason why you need to copy quite a few categories, which can take a lot of time . However copy/paste option is available for products only. Thus you can use category Import/Export Wizard to simplify this task.

Lets say you have 5 parent categories, each of them has to include 30 similar subcategories. You've entered all 30 in Category One. Make sure this is current(selected) category and click Import/Export button, select Export option. Specify the file you want your subcategories to be exported to (e.g. subcategories) and the path to it. Than select "Only from current category" option for on the next page and proceed with the export.

Then select all the subcategories, drug them to Category Two and drop there. Subategories will be moved to that category automatically. Now perform import of the file you've created (here we used "subcategories"). Category One will list the Subcategiries again. This way you'll have same subcategories in 2 categories.

Now Drag&Drop the subcategories From Category One into Category Three and import them again.

This trick can be used on basically any level of your category tree.