This link has some pretty good information on 'Perfect Fruits'. It looks like when you plant a 'Perfect fruit' tree you get a random amount of shakes before it dies. It would be pointless to NOT plant/shake the tree when you have them available. You should always have some perfect fruits on hand for when your 'Perfect Fruit' trees die.
Sidenote from the site: Perfect fruits sold in towns that the fruit is not native nets you 3000 bells (compared to the 600 in your town). So definitely sell your perfect fruits in your friends towns!
Note that you cannot plant perfect fruits of non-native species; if your fruit is orange, you cannot plant perfect peaches; they will instead turn into normal peach trees. However, if you imported a perfect orange into your town where oranges are native, you could plant that orange and it would become a perfect orange tree.