  While random forests are commonly used for regression problems, existing
methods often lack adaptability in complex situations or lose optimality under
simple, smooth scenarios. In this study, we introduce the adaptive split
balancing forest (ASBF), capable of learning tree representations from data
while simultaneously achieving minimax optimality under the Lipschitz class. To
exploit higher-order smoothness levels, we further propose a localized version
that attains the minimax rate under the H\"older class $\mathcal{H}^{q,\beta}$
for any $q\in\mathbb{N}$ and $\beta\in(0,1]$. Rather than relying on the
widely-used random feature selection, we consider a balanced modification to
existing approaches. Our results indicate that an over-reliance on auxiliary
randomness may compromise the approximation power of tree models, leading to
suboptimal results. Conversely, a less random, more balanced approach
demonstrates optimality. Additionally, we establish uniform upper bounds and
explore the application of random forests in average treatment effect
estimation problems. Through simulation studies and real-data applications, we
demonstrate the superior empirical performance of the proposed methods over
existing random forests.
