Affine invariance
Show that if the jth predictor is replaced by ax_j+b with a\ne 0, every candidate split on that variable has the same \Delta I, with the cut-point transformed to a\lambda+b. What changes if a<0? Does the same hold for a rotation of the predictors?
For a monotone increasing map x_j\mapsto ax_j+b with a>0, the set of observations with x_j<\lambda is the set with ax_j+b<a\lambda+b. Every split partitions the data identically and has the same child compositions, hence the same \Delta I. The best split is at the transformed cut. If a<0, the left and right children swap, and \Delta I is symmetric in the two children, with \hat p_L exchanged for 1-\hat p_L, so the reduction is unchanged. A rotation does not preserve this. A split on a single coordinate after rotation is a split on a linear combination before the rotation, which an axis-aligned tree cannot reproduce, so the trees and their accuracy can change. Training partitions are preserved by strictly monotone coordinate transformations, up to ties and equivalent cuts. Predictions between observed values can differ when a nonlinear transformation changes the midpoint convention.