@Reika:
The mythology on western phoenixes, like all mythological creatures, isn't consistent; sometimes they're reborn from the ashes of their former self, but at least in some old stuff I believe they just die and are reincarnated without all that fire fuss.
Additionally, this is almost certainly going to be a mix of eastern and western phoenix legends, which are two totally different beasts and I think only got conflated relatively recently (like eastern and western "dragons").
Of course, this is fantasy literature, not just an illustrated mythology, and is without doubt taking artistic license.