Generalizing the Comparison Sorting Lower Bound Proof
The decision tree method for proving lower-bounds (like the $\Omega(n \lg n)$ lower-bound for comparison based sorting you mentioned) are often called information theory methods.

Here is a short video by Faith Ellen about this method: 


Lower-bounds, Part 2/4: Information Theory


The first part is here:


Lower-bounds, Part 1/4: Introduction, Decision Trees