Usually making changes to a category tree should result in an immediate update on the front-end. However, if this is the first time you are working on your category tree, or if you are working on a new category structure that has to replace the existing one, there is one additional step to take. Go to the Magento Backend > Stores > All Stores. In this list, you can set the root category per website. So, you want to use a different category tree per website, that is also something you manage in this screen.