We prove the preservation theorems for NATP; many of them extend the previously established preservation results for other model-theoretic tree properties. Using them, we also furnish proper examples of NATP theories which are simultaneously TP₂ and SOP.

First, we show that NATP is preserved by the parametrization and sum of the theories of Fraïssé limits of Fraïssé classes satisfying strong amalgamation property.

Second, the preservation of NATP for two kinds of dense/co-dense expansions, i.e. the theories of lovely pairs and of $H$-structures for geometric theories, and dense/co-dense expansion on vector spaces is proved.

Next, we prove the preservation of NATP for the generic predicate expansion and the pair of an algebraically closed field and its distinguished subfield; for the latter, not only NATP, but also the preservation of NTP₁ and NTP₂ is considered.

Finally, we present some proper examples of NATP theories using the results proved in this paper, including the parametrization of DLO and the expansion of an algebraically closed field by adding a generic linear order. In particular, we show that the model companion of the theory of algebraically closed fields with circular orders (ACFO) is NATP.

我们证明了 NATP 的保存定理；其中许多扩展了先前建立的其他模型理论树属性的保存结果。使用它们，我们还提供了 NATP 理论的适当例子，这些理论同时是 TP ₂和 SOP。

首先，我们证明 NATP 通过满足强并合性质的 Fraïssé 类的 Fraïssé 极限理论的参数化和求和而得以保留。

其次，NATP对于两种稠密/共稠展开的保存，即可爱对理论和$H$-证明了几何理论的结构以及向量空间上的稠密/共稠扩展。

接下来，我们证明了NATP对于泛型谓词展开和代数闭域及其可区分子域对的保存性；对于后者，不仅考虑NATP，还考虑保留NTP ₁和NTP _{2 。}

最后，我们使用本文证明的结果提出了 NATP 理论的一些正确示例，包括 DLO 的参数化和通过添加通用线性阶来扩展代数闭域。特别是，我们证明了圆阶代数闭域理论（ACFO）的模型伴侣是 NATP。