It should recover the apps if you sync to your backup with the USB connection to your computer with iTunes - assuming that is where your backup is. If you are doing a iCloud backup and iTunes is not an option you can go to the App Store application on the iPhone, go to Updates and tap on the top of the screen "Purchased". You'll get a list of all your purchased apps and can reinstall them by clicking on the cloud.
The apps themselves are not part of the backup. If you backed up to your computer, then they should be reinstalled from iTunes as part of the post update process. (restore data, re-sync apps, re-sync media). If you backed up to iCloud, then they should be re-installed from the cloud.
If you backed up to your computer and the apps were NOT in your iTunes library, you will have to re-download them.
Did you make sure to use the "Transfer purchases function" in iTunes in order to transfer the iPhone apps to your PC before restoring the iPhone? If yes, the app files themselves (which are not part of the backup) are found in your iTunes library - here's how to make sure the iPhone apps are backed up to iTunes: How to transfer iPhone apps to PC