Why do we say "You are barking up the wrong tree"?

Meaning:

You are taking the wrong course and/or wasting your effort in the wrong direction

Background:

This expression alludes to a dog who believes it has chased its prey up one tree, when in fact the prey is somewhere else entirely.

The earliest known references both date to 1832: Westward Ho! by James Kirke Paulding and Legends of the West by James Hall. Most sources put Westward Ho! as the earlier, with some citing Legends of the West as publishing in 1833

"Here he made a note in his book, and I begun to smoke him for one of those fellows that drive a sort of a trade of making books about old Kentuck and the western country: so I thought I'd set him barking up the wrong tree a little, and I told him some stories that were enough to set the Mississippi a-fire; but he put them all down in his book." - James Kirke Paulding, Westward Ho!

"It doesn't take a Philadelphia lawyer to tell that the man who serves the master one day, and the enemy six, has just six chances out of seven to go to the devil. You are barking up the wrong tree, Johnson." - James Hall, Legends of the West

James Paulding was a close friend of Washington Irving and held a number of positions in the U.S. Navy including Secretary of the Navy for about three years. During his tenure he opposed the introduction of steam propelled warships stating that he would "never consent to let our old ships perish, and transform our Navy into a fleet of sea monsters."

James Hall was a United States judge and writer who has been called a literary pioneer of the Midwestern United States.

Irrespective of who coined the expressions, Hall, Paulding or an unknown body preceding them, the expression appears to have caught on in this period as it can be found in a number of American newspapers during the 1830s.

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