over the leaves, and such that each $\bx_{i} \in \mathbb{S}$ belongs to exactly one leaf $L_{j}$. Thus, the whole model class $\mathcal{F}$ can be represented as follows: $$\begin{aligned}
\label{eq:class_str}
\mathcal{F} = \bigcup_{\substack{T \in \cT_l\\(i_{s},t_{s})_{s} \in ([D] \times \mbr)^{\cT^\circ}}}\mathcal{F}_{T,(i_{s},t_{s})_{s\in T^{\circ}}} ,\end{aligned}$$ where we formally write $\mathcal{F}_{T,(i_{s},t_{s})_{s\in \cT^\circ}} = \{f: f = (f_{1},\ldots,f_{\ell}): \forall x \in \mathcal{X},  f(x) = \sum_{j} \inner{f_{j},x}\mathbbm{1}(x \in L_{j}), f_{j} \in \partial T \} $ for each tree $T$ and split family $\paren{i_{s},t_{s}}_{s \in T^{\circ}}$. To study the statistical performance of the classes with the union-type structure we make use the of Lemma 2 from [@maurer2014inequality] and its corollary, which we provide below for completeness.

\[thm:sup\_bound\] Let $N \geq 4$ be some natural number and $A_{1},\ldots,A_{N} \subset \mathbb{R}^{n}$ some subsets, such that for a given $A \subset \mathbb{R}^{n}$ we have $A =\bigcup_{i=1}^{N} A_{i}$. Consider $\bm{\sigma} = \paren{\sigma_1,\ldots,\sigma_n}$ to be a vector of i.i.d. Rademacher variables (i.e. uniformly distributed over $\set{-1,1}^n$). Then we have: $$\begin{aligned}
    \ee{}{\sup_{z \in A} \inner{\bm{\sigma},z}} \leq \max_{i=1}^{N}\ee{}{\sup_{z \in A_{i}}\inner{\bm{\sigma},z}} + 4 \sup_{z \in A}\norm{z}\sqrt{\log{N}}.
    \end{aligned}$$

Assume we have a finite family of functional classes $\mathcal{F},\mathcal{F}_{1},\ldots,\mathcal{F}_{N}$, such that $\mathcal{F} = \bigcup_{i=1,\ldots,N} \mathcal{F}_{i}$ and a sample $\mathbb{S}$ as before. Denote $A_{j} \subset \mathbb{R}^{n}, A_{j} = \{\paren{f(\bx_{1}),\ldots,f(\bx_{n})}: f 