I'm studying trees for the first time and I got stuck on a theoretical passage: on ISLR (Hastie, James, Tibshirani, Witten) I found that considering that: $$\sum_{j=1}^{J} \sum_{i\in Rj} (y-\hat{y}_{R_{j}})^2 $$ minimizes the RSS, this formula would be the best one for finding non-overlapping regions. However it has been stated that such an approach would be np-hard, hence a greedy-approach should be preferred.

My point is: how exactly do we find the regions? Considering that I have $$ X_{1}, X_{2},...,X_{p} $$ predictors, how to split the space into $$ R_{1}, R_{2},...,R_{j} $$ regions? What's the theoretical intuition?


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