The carpenter legally changed his name to Woody Carpenter as a gimmick.
Inverted: Woody Carpenter is allergic to wood.
Subverted: Woody Carpenter is a baker.
Double Subverted: ...but his favorite hobby is building birdhouses.
Parodied: Woody's brothers are Ash and Rowan, his sister is Willow. Their parents were both killed in a freak lathing accident and they were raised by old Ma Hogany. When Woody is bitten by a radioactive carpenter ant and given elemental powers over wood, his first thought is to call himself... Mister Amazing!
Zig Zagged: The Woodsman's backstory changes Depending on the Writer. Sometimes he was born Woody Carpenter, sometimes Jack Lumber. Sometimes he's Steve Bunyan, revealed to be a descendant of Paul Bunyan. Sometimes he has an entirely mundane name with nothing to do with wood. These identities are regularly Ret Conned.
Averted: The carpenter's name is Steve Smith, or something equally normal.