A lot of the documentation for recovering folders from exchange says that you can do it straight from outlook by selecting the recover deleted items icon Outlook.
In theory this is fine but in practice your exchange users do not have permission to do this and they will get an error saying:
Outlook was unable to recover some or all of the items in this folder. Make sure you have the required permissions to recover items in this folder and try again. If the problem persists contact your administrator.
Your public folder that has been deleted will still be held by exchange because exchange has a retention policy so when a folder is deleted it is still held in the exchange retention folder.
Recover Public Folders using Powershell
To restore the public folder you will need to open an exchange PowerShell if you are using Office 365 follow the steps in our post how to connect to office 365 using powers shell to connect to Office 365 using PowerShell.
Then you need to list all of the public folders and output to CSV by running the command(you will need to set the output of Export-Csv to the folderpath of the file you want to output):
Get-PublicFolder -Identity "\NON_IPM_SUBTREE" -Recurse | Select Identity | Export-Csv c:\outfolder\out.csv
Then open up the csv file and find the folders you would like to restore then go back to powershell and you will need to run the following command replacing the *VALUE* the the value from the CSV file.
Set-PublicFolder -Identity "*VALUE*" -Path "\" -Verbose
Set-PublicFolder -Identity "\NON_IPM_SUBTREE\DUMPSTER_ROOT\DUMPSTER_EXTEND\RESERVED_1\RESERVED_1\a33b5631-a4a8-432f- 9349-80aa14eadbf8\2015\General Correspondence" -Path "\" -Verbose
You might also like this article about how to recover a public folder database which will allow you to recover the public folders after the retention period on an on premis installation of Exchange Server.