a discussion, and proofs are included in an Appendix.

[0.32]{} ![Decision sets for different methods. Red crosses denote minority class samples, while blue points denote majority class samples. Rectangles with dashed frames denote the support of minority class samples, while rectangles filled with green color denote the minority class decision set.[]{data-label=""}](toy_CART_unpruned.png "fig:"){width="\textwidth"}

[0.32]{} ![Decision sets for different methods. Red crosses denote minority class samples, while blue points denote majority class samples. Rectangles with dashed frames denote the support of minority class samples, while rectangles filled with green color denote the minority class decision set.[]{data-label=""}](toy_CART_pruned.png "fig:"){width="\textwidth"}

[0.32]{} ![Decision sets for different methods. Red crosses denote minority class samples, while blue points denote majority class samples. Rectangles with dashed frames denote the support of minority class samples, while rectangles filled with green color denote the minority class decision set.[]{data-label=""}](toy_SVRTree.png "fig:"){width="\textwidth"}

Methodology
===========

We first introduce the definition of surface-to-volume ratio (SVR) and tree impurity, and then define SVR-Tree as the minimizer of a weighted average of tree impurity and SVR. We then state the algorithm to estimate this tree from training data $\mathscr{D}_n = \{(X_i, Y_i)\}_{i=1}^n$. We assume readers have basic knowledge of tree-based classifiers like CART [@breiman1984classification] and C4.5 [@quinlan2014c4]. In 