# How are the Average treatment effects using Causal Trees (CT) and Causal Forest (CF) decision-trees estimated?

I have read the following paper: https://iopscience.iop.org/article/10.1088/1748-9326/aafa8f/pdf . The paper seem to be understandable, however, several questions I have got about causal trees.

As I understand conditional average treatment effect (CATE) shows the average increase in the dependent variable (Y) for the sample? What is the difference between the results of the CATE obtained for the full sample and only treatment group?

Moreover, in the paper, the authors told that they used grf package, however, in grf the function "causalTree" only computes Random Forests trees, but not the simple (pruned) Causal Tree. As a result, how did authors get the Figure 2? Additionally, if one decided to use Random Forest, it is necessary to tune some hyper-parameters such as number of variables to be selected at each node, but in the causalTree function is impossible to do that. As a result, can we say that this function is incomplete in terms of availability of tuning options?

Furthermore, causalTree function allows automatically tune hyper-parameter alpha via cross-validation, but how to get this cross validation plot, where on y-axis is Mean Squared Error and on x-axis is alpha?

Finally, on p.7 authors said that "corresponding to minimum cross-validation error to maximise predictive power without ‘overfitting’ (size of tree=15; table S2(a), figure S1) [18]". However, where this table and figure can be located?

Sorry, for such a large number of question. I am a bit new to the estimation of the treatment effects in R, but want to fill gap in the understanding of this paper.