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CARTs naturally consider interactions between covariates, see:

Do CART trees capture interactions among predictors?

I would like to know whether it is always possible to translate the variable interactions when converting a classification and regression tree into a generalised linear model.

For example, the interaction between economic conditions ($X_1$) and type of building purchased ($X_2$) can be given by a regression specified with Wilkinson notation:

$$Y \sim X_1:X_2$$

That is, find the contribution to the response based on all possible values of the cartesian product of $X_1$ and $X_2$.

Is there a general rule that can be applied to translate an arbitrary interaction in a CART?

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Tree algorithm itself assumes the interactions so you do not need to specify the product term directly in your model equations, but in linear regression for example it is is different and you need to specify the product term.

The interpretations of interactions depends on the splits the tree is forming and applies locally to the group which fall in those splits... if I've understood you correctly.

There's a similar question also here:

How to include interaction terms in R/tree model?

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