I'm a user of ctree function from partykit package in R. I always wondered for which purpose we want to use Monte Carlo to compute the distribution of test statistics? The literature suggest that it goes for large samples. I checked vignettes for partykit and coin but I didn't get clear answer. Are there any different conditions we might use it? I noticed that when the MC is used, ctree returns more variables for a given tree. Why?
The default method for obtaining the p-values in conditional inference trees is by using the asymptotic permutation distribution (which is normally or chi-squared distributed, depending on the type of test). Alternatively, you can approximate the exact finite-sample permutation distribution by drawing a sufficiently large number of Monte Carlo samples. The two approaches will typically be similar - increasingly so for larger sample sizes. In small(er) samples there might be a bit of a difference with the approximated finite-sample distribution sometimes leading to slightly more powerful tests. Hence, in your case I suspect that the sample size is small to moderate and you get a few more significant splits in the tree.