Will Time Machine overwrite files intentionally deleted on my remote hard drive?

I want Time Machine to overwrite files intentionally deleted on my remote hard drive.

I've used a 2TB remote hard drive exclusively for Time Machine backups of two MacBook Pros (1 and 2). They filled the remote hard drive to about 90% of capacity. MacBook Pro 2 has been given to my son (who has his own backup drive) so I've "deleted" the backup folder and its files for MacBook Pro 2, and continue to back up MacBook Pro 1 plus a new MacBook Air.

MacBook Pro 2 was left connected to the remote hard drive for about 48 hours and set to "empty trash," but the pace would have taken a very very long time to fully erase deleted files.

If I do nothing further, will Time Machine overwrite the MacBook Pro 2's backup files on the remote hard drive? If not, what is the solution to enable either overwriting or speedy erasure of files already marked deleted?