$c_3>0$, $\beta\in (0, 1/2)$, we have $$\lim_{n\to\infty}\mathbb{P}(\hat{T}_n \;\text{does not have partitions in}\; X[j_{q+1}], X[j_{q+2}], \cdots X[j_d]) = 1.$$

In Theorem \[feature selection\] conditions 1 and 2 are complementary. If $c_2$ is smaller (i.e., condition 2 is stronger), $c_1$ can be smaller (i.e., condition 1 is weaker). The opposite also holds. The following two Corollaries cover special cases.

\[feature selection monotonic\] If the optional steps in Algorithm 1 are enabled, condition \[condition 1\] is satisfied with $c_1=1$ (that is, in each hyper-rectangle, the maximal impurity decrease at non-redundant features is equal to the oracle impurity decrease) and $\lambda_n\ge c_3 n^{-(1/2-\beta)}$ for some constant $c_3>0$, $\beta\in (0, 1/2)$, we have $$\lim_{n\to\infty}\mathbb{P}(\hat{T}_n \;\text{does not have partitions in}\; X[j_{q+1]}, X[j_{q+2}], \cdots X[j_d]) = 1.$$

\[feature selection independent\] If the optional steps in Algorithm 1 are enabled, $X'$ and $X''$ are independent and $\lambda_n\ge c_3 n^{-(1/2-\beta)}$ for some constant $c_3>0$, $\beta\in (0, 1/2)$, we have $$\lim_{n\to\infty}\mathbb{P}(\hat{T}_n \;\text{does not have partitions in}\; X[j_{q+1]}, X[j_{q+2}], \cdots X[j_d]) = 1.$$

Corollary \[feature selection independent\] is a direct result of theorem \[feature selection\] with $c_2=0$ and an arbitrary value of $c_1$.

Numerical Studies
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We compare SVR Tree with popular imbalanced classification methods on real datasets, adding redundant features 