"Killing a swan is treason because the Crown owns all swans."
The Crown does hold the right of swan ownership on certain stretches of the Thames, but killing one has never been an act of treason. The Law Commission's Statute Law Repeals team published "Legal Curiosities: Fact or Fable?" in March 2013 and traced this claim to its origin: a confusion between the Crown's ceremonial right of ownership and the actual statutory offence of treason, which has never included swans.
The treason statutes (Treason Act 1351, Treason Act 1702, and others) list the acts that constitute treason. Killing a bird is not among them. The claim is a myth, and the debunk is better content than the myth ever was.