GitHub Pages has 2 different types of sites: User/Organization sites and Project sites:
User/Organization sites are created when you name the repository
username.github.io and can only be served from the master branch. From the looks of your screenshot it seems like you’ve created this type of site, meaning the dropdown menu isn’t available.
If you’d prefer to serve your site from the
gh-pages branch then you’ll need to change the type of Pages site to a Project site by renaming your repository to something other than