Not the most famous western directed by John Sturges. But the 'search for hidden money/gold' theme usually makes a film more appealing to the audience.
I think you're right on this one, MI6; not a great Sturges film, but a pretty damn good one. The funny part is, I don't remember the tree in it. I'm basing my guess on the costume of the foreground character.
According to an 29 April 1958 Hollywood Reporter news item, THE LAW AND JAKE WADE marked the first time a major film studio used a "canned soundtrack" for background music. The item stated that composer Bronislau Kaper had begun a score for the film, but due to the musicians' strike, M-G-M purchased a full-length recording from the library of Capitol Records and used it for the film's music track.