The $\alpha$ rules are different from the $\beta$ rules. For the $\alpha$ rules, you create a new branch for each of $\alpha_1$ and $\alpha_2$. But for the $\beta$ rules, you don't branch. Rather, you put both $\beta_1$ and $\beta_2$ in the same branch.
If you think about the rules conceptually, and what you are trying to do with this tableaux method this should all makes sense. That is, in general, in a semantic tableaux method you exolore all possible options to make some statement true. Thus, for example, there are two ways to make a statement like $P \lor Q$ true: either $P$ is true, or $Q$ is true. Hence, you create a branch for each of those ways. n the other hand, there is only one way to make $\
eg (P \lor Q)$ true, and that is by setting both $p$ and $Q$ to false. This is why you get both $\
eg P$ and $\
eg Q$ still in the same branch.
Below is a pic of the tree as it should look like:
![enter image description here](