The “tree” and the “cat” are related in the general sense of asking what role perception or observation plays in reality, and they suggest opposite answers.
The case of the tree is meant to cast doubt on our natural experience–that the tree definitely makes a sound when it falls. Does the falling tree really “make” sound though, or is sound merely the “interpretation” our minds impose on the experience of our ears picking up a “disturbance” in the air? (You can raise a similar question for any of our senses.)
The case of the cat was in fact posited by Schrödinger as a “ridiculous” (his word) example of quantum mechanics, because he wanted to highlight what he considered (and many physicists still struggle with) a grave conceptual problem with the theory, namely how does the microscopic indeterminacy of reality transform itself into the macroscopic certainties we perceive? He regarded the supposed critical role of observation as hardly satisfactory since the cat in the still-closed box had to be alive or dead (or dying), no “two ways” about it.
