Deleted I tunes Backup file I deleted my most current I Tunes back-up file, it had my daughter's BDay photos from last night. I have been on the phone with an Apple agent and no luck.  It seems strange to me that this file would not be in the trash bin or somehow be able to be extracted.  Where could it have gone?  Is there someone out there or a product that can help me locate this back up? BTW, I thought it was an older back-up so I deleted it. (the name confused me- Iphone 3 rather than Iphone 5).  They are not on iCloud and I do not use Time machine, so my only hope is to recover from my iMac.

Thanks
You can get one of many data recovery softwares. They are not free. It doesn't matter what kind of content you lost, so recovering the iTunes backup is the same process as recovering a text document, etc.

First of all, stop using the computer (or that particular hard drive). The more you use it, the more files are written to disk, and you lower your chances of being able recover the file.
I have had good experiences with Data Rescue in the past. (http://prosofteng.com).

If the deleted file was on the same drive as your OS (the boot drive), you will need to use a different system for the recovery software. You can download and install your recovery software of choice on a different Mac, then boot your iMac in Target Disk Mode and connect them via FireWire. (http://support.apple.com/kb/ht1661)

If the deleted file was on an external drive, you don't need a second Mac; just run the recovery software. In any case, you will need to know the location the file was deleted from. Normally that would be ~/Library/Application Support/MobileSync/Backup/