I think I remember seeing somewhere in the Talmud that the King's haircut entailed that each hair should be exactly the same length.
Am I mistaken, or does anyone know the source?
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The every hair being the same length might be the haircut of a כהן גדול, as referenced in Joel's answer, from Sanhedrin 22b. That gemara also discusses a King's haircut and says that he has a haircut every day in order that he always look respectable, though the gemara does not mention the length of that haircut.