$T$ divides the sample space $\Omega \subset \mathbb{R}^d$ into two disjoint subsets $\Omega_0$, $\Omega_1$, where $\Omega_0 \cup \Omega_1= \Omega$. The tree $T$ predicts a new sample $X^*$ belongs to class 0 if $X^* \in \Omega_0$, and class 1 if $X^* \in \Omega_1$. The surface-to-volume ratio of a classification tree is defined as the surface-to-volume-ratio of set $\Omega_1$: $r(T) = r(\Omega_1).$

Impurity Function and Tree Impurity
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A classification tree partitions the sample space into multiple leaf nodes, assigning one class label to each leaf node. The tree should be built to maximize homogeneity of the training sample class labels within nodes. This motivates the following definition.

An impurity function $\phi(\cdot, \cdot)$ is defined on the set of pairs $(p_0, p_1)$ satisfying $p_0\ge0, p_1\ge0$, $p_0+p_1=1$ with the properties (i) $\phi(\cdot, \cdot)$ achieves its maximum only at $(1/2, 1/2)$; (ii) $\phi(\cdot, \cdot)$ achieves its minimum only at $(1, 0), (0,1)$; (iii) $\phi(\cdot, \cdot)$ is symmetrical in $(p_0, p_1)$, i.e., $\phi(p_0, p_1) = \phi(p_1, p_0)$.

Let $p_0$ and $p_1$ represent the probabilities of belonging to the majority and minority class, respectively, within some branch of the tree. Ideally, splits of the tree are chosen so that, after splitting, $p_0$ and $p_1$ move closer 