Are you certain that at some point the branches weren't painted brown? Perhaps the paint from the fruit was removed by frequent dishwashing and the brown is all that remains? I might try a little fingernail polish remover on a q-tip and see if that removes it (or paint remover), just to be sure if it's paint or not.

Greased Lightning; test first. I used it to clean years of coal dust and smoke off of vintage pottery and glass.

I'd try rubbing alcohol. It's pretty good on greasy/sticky grime

Thanks, I'll try these.

Olychick, it's definitely not paint. Since I have my Mom's plates that match these, hers don't have the brown residue.

I don't know what has stained your plates, but I've used household bleach to remove tea/tannin stains from vintage tea cups. Works perfectly. Swab it on with a paper towel, then wash, rinse thoroughly. Keep in mind that is for a stain, not a build up of actual food remnants from long ago.