For AVL Trees why is keeping a trit (left heavy, right heavy or balanced) sufficient?
The core argument is that the balance can never change by more than one when inserting or deleting a single key; of course, since heights can only change by at most one.

That way, knowing the old balance (which is in $\{-1,0,1\}$) and if (and how) subtree heights changed is enough to know the new balance (which is in $\{-2,1,0,1,2\}$).
If the balance is $-2$ or $2$ we rotate, of course, and restore balance (in that node).

In order to prove this, simply write out all possible cases of old balances and changes due to insertion or deletion.