Recursive decision trees are widely used to estimate heterogeneous causal treatment effects in experimental and observational studies. These methods are typically implemented using CART-type recursive partitioning and are often viewed as adaptive procedures capable of discovering treatment effect heterogeneity in high-dimensional settings. We study causal tree estimators based on adaptive recursive partitioning and establish lower bounds on their estimation accuracy. Under basic conditions, we show that causal trees constructed via standard CART-type splitting rules cannot achieve polynomial-in-$n$ convergence rates in the uniform norm (where $n$ denotes the sample size). The underlying mechanism is that greedy recursive partitioning selects highly imbalanced splits with non-vanishing probability, producing terminal nodes containing very few observations and leading to large estimation variance. We further show that sample splitting (``honesty'') yields at most negligible improvements in convergence rates. As a consequence, causal tree estimators may converge arbitrarily slowly and can even be inconsistent in some settings. Our results also clarify the role of balanced partition assumptions in existing theoretical guarantees for causal forests and related ensemble methods. The analysis develops new probabilistic tools for studying adaptive recursive partitioning procedures, including non-asymptotic approximations for suprema of partial sums and Gaussian processes. As a technical by-product, we also identify and correct an error in Eicker (1979).

翻译：递归决策树被广泛用于估计实验与观测研究中的异质因果处理效应。这些方法通常通过CART型递归划分实现，常被视为能够发现高维情境下处理效应异质性的自适应过程。我们研究了基于自适应递归划分的因果树估计量，并建立了其估计精度的下界。在基本条件下，我们证明通过标准CART型分割规则构建的因果树无法在一致范数中达到关于样本量$n$的多项式收敛速率。其内在机制在于：贪婪递归划分会以非零概率选择高度不平衡的分割，产生包含极少观测值的终端节点，从而导致较大的估计方差。我们进一步证明样本分割（"诚实性"）至多只能带来收敛速率的微不足道的改进。因此，因果树估计量的收敛速度可能任意缓慢，在某些设定下甚至可能不具备一致性。我们的结果也阐明了平衡划分假设在现有因果森林及相关集成方法理论保证中的作用。该分析为研究自适应递归划分过程开发了新的概率工具，包括部分和与高斯过程上确界的非渐近逼近。作为技术副产品，我们还发现并修正了Eicker（1979）中的一处错误。