HELP! My Time Machine deleted old backups

You can't. You need a larger backup drive, probably at least twice the size of what you have. You also need at least two independent backups to different destinations.

The problem is that Time Machine is only a temporary repository. It will always ensure that you can restore the machine to the state that it was in at the last backup. But if the backup drive runs out of space - as yours undoubedly did when you installed a brand-new system, forcing Time Machine to back up everything on the entire drive all over again - then it will start deleting older files until it makes enough room to back up the newer files.

So, if you only had one backup, with Time Machine, and it wasn't large enough (at least double what you planned on backing up) ,and you didn't restore all your files before letting Time Machine start backing up the new system, there is unfortunately nothing you can do at this point. Your data is not only gone, but it has almost certainly been written over. You could always try a data recovery tool on the backup drive, but results are likely to be poor. Some such tools are DataRescue, FileSalvage and Stellar Phoenix.

In the future, you need to:

• ensure that the Time Machine backup drive has a minimum of 2 times the capacity of what you plan to back up, now and in the foreseeable future
• make at least one other backup on another drive, preferably more than one (minimum of two backups total)
• use a different backup program, such as Carbon Copy Cloner, for at least one of those backups
• keep at least one backup off-site - but frequently updated - in case of fire, flood, theft, etc

  (I keep Carbon Copy Cloner backups on two drives, one of which is always in a safe deposit box at the bank, and swap them out about once per month.)