There's a dead, gnarly old tree in the courtyard outside of Denethor's 'throne room' on the top level of Minas Tirith. Gandalf told Pippin that it's called the White Tree of Gondor. We see it several times in 'Return of the King' and it's obviously dried-up dead. Yet after they defeat Sauron and Aragorn is crowned King just weeks later, it's suddenly flowered and in full bloom, blowing its petals across the coronation. How can that be?
One word: Jackson. A second word: nonsense.
The story is different in the books - Aragorn finds a sapling of the dead tree and plants it, which becomes the new White Tree:
Then Aragorn turned, and there was a stony slope behind him running down from the skirts of the snow; and as he looked he was aware that alone there in the waste a growing thing stood. And he climbed to it, and saw that out of the very edge of the snow there sprang a sapling tree no more than three foot high. Already it had put forth young leaves long and shapely, dark above and silver beneath, and upon its slender crown it bore one small cluster of flowers whose white petals shone like the sunlit snow.
This ain't no normal tree either:
Verily this is a sapling of the line of Nimloth the fair; and that was a seedling of Galathilion, and that a fruit of Telperion of many names, Eldest of Trees.
And Telperion, as one of the Two Trees of Valinor, the light of which was captured in the Silmarils, is therefore quite central to all of Tolkien's work.
(Quotes from RotK Book VI Chapter 5)
Indeed, Peter Jackson did somewhat over-dramatise this part, though I find the original fairly dramatic/moving in of itself.
Further to Jimmy S's point, Tolkien describes the history of Minas Tirith, where the White Tree would regularly start to die but the King/Stewards would plant a new sapling:
When King Telemar died the White Tree of Minas Anor also withered and died. But Tarandor, his nephew. who succeeded him, replanted a seedling in the citadel.
[LotR appendix A., I(iv)]
The first sapling was planted by Isildur (before he lost the ring), when the city was still Minas Anor. [LotR Book II, ch. 2, The Council of Elrond]
Also, the finding of the sapling in the snow was a prophecy linked to the return of the king, and the fact the the White Tree died without a sapling was indicative (allegorically) of the petering out of the line of Kings/Stewards:
When Belecthor II, the twenty-first Steward died, the White Tree died also in Minas Tirith, but it was left standing 'until the King returns', for no seedling could be found.
[LotR Appendix A., I(iv) The Stewards]
As an aside, yes Sauron wasn't as powerful as Morgoth, but that's because he was only his "lieutenant" - he was a Maia like Gandalf. He still fooled the Elves into making his rings and engineered the destruction of Númenor though!
Others have answered in terms of the books, but another answer is possible if we ignore the books (I can't believe I just said those words!), and look only for movie-logic
Basically the answer is, "it's magic". But it's not completely abrupt. There's a moment in the movie (when Denethor is en route to immolating himself) when we see the tree in the foreground, and just for a moment the camera lingers on a single flower that has appeared on the "dead" tree!
It's also not quite as lame as "it's magic" might suggest. I see it as a kind of "fisher-king" thing: the life of the tree is tied to the kingship; as Aragorn's day approaches, the dead (or at the very least, extremely dormant) tree begins to awaken.