How can decision trees be tuned for non-symmetrical loss? Suppose we use a decision tree to predict if a bank customer can pay back a credit. So it is a two class classification problem. Now we can make two mistakes:


$\alpha$ error: The customer can back the credit, but we predict he can't.
$\beta$ error: The customer can't pay back the credit, but we predict he can.


Now we know that $\beta$ errors are 123.4 times as expensive as $\alpha$ errors. But we only have a given set of data. In this set we have $n_1=10000$ customers who paid back the credit and $n_2 = 100$ customers who didn't.

How can the training of the decision tree be adjusted to account for the fact that $\beta$ errors are more expensive?

(Note: This is a theoretical question to learn about decision trees. I know about other classifiers like neural networks and I know of ensembles. However, I only want to know about decision trees here.)
Theoretically, decision tree algorithms specify the feature as well as the threshold that maximize the separation between classes at each node. This can be done by solving optimization problem related to the entropy at the children nodes. 
You can modify this optimization problem by including the miss- classification costs you have in the optimization problem so the algorithm will be biased toward the most expensive class