By default, Windows Operating System will show 2 icons from Desktop only and both of them are My Computer(Computer in Windows Vista and above) and Recycle Bin, somehow you might discover the Recycle Bin disappear after using for some times.
To restore and remove the recycle bin button, it just a reverse way as step below.
First of all, you will need to check if your desktop icons hidden from your setting.
Second, you will need to check your Desktop Icon Setting.
1. Right click from anywhere of your Desktop.
2. Select on the Personalize.
3. Select on Change Desktop Icons from the left panel.
4. You shall be able to see image as above and select on the Recycle Bin radio button to enable it and remove it to disable it.