The truth tree generator https://www.umsu.de/trees/#~6x~7yAxy~5~7y~6xAxy is providing the countermodel
Domain: { 0, 1 }
A: { (0,0), (1,1) }
to show that $$∀x∃yAxy → ∃y∀xAxy$$ is an invalid sentence.
I don't understand how this is a countermodel that shows that the formula is invalid. Can someone help me understand why, please?
The given set in the answer is the set where predicate $A$ is true.
Otherwise said, the countermodel can be expressed in the following way
$$\begin{cases}A00, \ A11 \ \text{are true} ;\\A01, \ A10 \ \text{are false} \end{cases}.$$
In this context, the left hand side formula is valid, whereas the right hand side is not because no $y$ satisfies it, indeed:
taking $y=0$, you cannot say $\forall x, Ax0$ because it is false for $x=1$,
taking $y=1$, you cannot say $\forall x, Ax1$ because it is false for $x=0$.