There's an exercise question in my book (Formal Logic: Its Scope and Limits by Richard Jeffrey) that asks whether or not $\forall x (\exists y Lxy \rightarrow \forall y Lyx)$ is a logical truth.
How would i set this up on the tree? My first thought was to compare this with a negated version of the statement, but that would be trivial, since it would be logically equivalent to one another anyway?
Yes, you need to negate it for the setup. Then, if all branches close, it is a tautology, and if you get a finished and open branch, it is not a tautology.