No... putting it in the trash will only erase the application.... but everyone knows that.
I don't know the version of Mac OS X you have and I don't remember everything prior to X.2 .
With Jaguar, when you install an application, you will find a log file of the program you have installed in your root, somehting like "Apple Remote Desktop Installer log file" or "ADR Installer log file". In it you will find every compononent installed in your system.
If I were you I'd install ARD again and search for that log file. Since you already have it installed it won't hurt to re-install it.
Hope this helps.