events that include $n$ random variables $\{(X_i, Y_i)\}_{i=1}^n$, we simply use $\mathbb{P}$ to represent the probability measure in the $n$-product space. For example, for $\mu>0$, $\mathbb{P}\left((\sum_{i=1}^n X_i)/n >\mu\right)$ denotes the probability of the event $\{\{(X_i, Y_i)\}_{i=1}: (\sum_{i=1}^n X_i)/n > \mu\}$. We use $\mathbb{E}$ for expectations over $\mathbb{P}$ and $\mathbb{E}_n$ for expectations over $\mathbb{P}_n$.

Surface-to-Volume Ratio
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For all $d \in \mathbb{N}$, define a $d$-dimensional measure space as $(\mathbb{R}^d, \mathscr{B}, \mu)$, where $\mathscr{B}$ is the collection of Borel sets, and $\mu$ is Lebesgue measure. For any measurable closed set $A \in \mathscr{B}$, we define its volume as Lebesgue measure of set $A$: $V(A) = \mu(A)$. We define the surface as the $d-1$ dimensional Lebesgue measure of the boundary of $A$: $S(A) = \mu_{d-1}(\partial A).$ For any set $A$ with $0<\mu(A)<\infty$, the surface-to-volume ratio (SVR) can be obtained as $r(A) = \frac{S(A)}{V(A)}.$ For sets with the same volume, the $d$-dimensional ball has the smallest SVR, while sets having multiple disconnected subsets and/or irregular boundaries have relatively high SVR.

#### Surface-to-Volume Ratio of a Classification Tree

For training data $\mathscr{D}_n$, we define a closed bounded sample space $\Omega\subset \mathbb{R}^d$ such that the support of $X$ is a subset of $\Omega$. A classification tree 