I'm not sure why Robin's answer was not accepted, but it has indeed been long confirmed that Devil Fruits re-appear after the user dies. On the other hand, whether artificial Zoan fruits re-appear has not been confirmed at the time of writing.
In chapter 676, it was the first time we could see a fruit re-spawn. When Smiley dies, we can see the nearest apple change form into the Salamander fruit, model Axolotl, which was the fruit that was injected into the poisonous H2S gas, creating Smiley. It seems like at least all pirates related to Donquixote Doflamingo know about this process, as Caesar put a bag of apples on the sledge on purpose, commented that he will meet Smiley again.
In chapter 703, it was then finally confirmed that Devil Fruits do indeed re-spawn after a user dies. When Luffy and Franky were interrogating one of Doflamingo's underlings, he revealed that the Fire Fruit would be put up as the price for the Colosseum contest. He then confirmed that a fruit can exist only once and that they are indeed reborn, making this Fire Fruit, Ace's Fire Fruit.
The Devil's fruit... It is said two of the same kind can't exist at the same time. However... It appears that when an ability user dies, that Devil's fruit is reborn somewhere in this earth! Fire fist Ace's Fire Fruit was unknowingly, brought back to this world after his death... And is now in possession of the young master!