If you knock out Azelf in Pokemon Diamond, Azelf will be gone forever (unless you have saved before battling Azelf and shut off the game and turn it back on to try again).
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When you knock out a roaming Pokemon, it will faint and then start roaming again, but with its health at full.
You cannot re-battle Azelf once beating it if it's in Pokémon Diamond or Pokémon Pearl however if you knock it out in Pokémon Platinum, Pokémon Black 2 or Pokémon White 2 then you will be able to re-battle it. As of Pokémon Platinum, there was a new re-spawning feature that was implemented for Legendary Pokémon that will allow you to re-spawn them if you have knocked them out, all you have to do is head to the Pokémon League and beat the Champion.
You cannot regenerate Legendary Pokémon in Pokémon Diamond. The regeneration feature for Legendary Pokémon was created as of Pokémon Platinum so it is not available in Diamond so for Legendary Pokémon in Diamond, you will have to get any of the Legendaries that you have knocked out from another game however you can save before challenging the Pokémon so if you knock it out, you can just turn off your DS and turn it back on so it'd be before you challenged the Pokémon.
i am not sure but you have to knock treeis with headbut attack
Yes, you get it back righr after the battle.