The tree tables are testing for satisfiability, and in this case we are testing if two formulas α and β are equivalent. The negated biconditional ¬(α ↔ β) represents a contradiction if the two formulas are equivalent, therefore all paths will close on the tree.
However, if the two formulas are equivalent, would doing a tree beginning with α and ¬β, one on top of the other, not also be a contradiction?
Why would this not work as a test for equivalence?