We give definitions that distinguish between two notions of indiscernibility for a set $\{a_\eta \mid \eta \in \W\}$ that saw original use in \cite{sh90}, which we name \textit{$\s$-} and \textit{$\n$-indiscernibility}. Using these definitions and detailed proofs, we prove $\s$- and $\n$-modeling theorems and give applications of these theorems. In particular, we verify a step in the argument that TP is equivalent to TP$_1$ or TP$_2$ that has not seen explication in the literature. In the Appendix, we exposit the proofs of \citep[{App. 2.6, 2.7}]{sh90}, expanding on the details.Comment: submitte

Abstract. We show that if T is any geometric theory having NTP 2 then the corresponding theories of lovely pairs of models of T and of H-structures associated to T also have NTP 2 . We also prove that if T is strong then the same two expansions of T are also strong.

We prove that the NTP 1 property of a geometric theory T is inherited by theories of lovely pairs and H-structures associated to T . We also provide a class of examples of nonsimple geometric NTP 1 theories. IntroductionOne theme of research in model theory is to inquire whether some well-known properties are preserved under a certain unary predicate expansions of a given structure. One of the motivations for this is that positive theorems of this kind often allows us to obtain interesting and complicatedlooking theories which still satisfy some strong tameness conditions. The study of expansions by unary predicates reaches back to the paper of Poizat on beautiful pairs [10], and has been developed in various directions ever since. Remarkable papers on this subject include for example [5], where the stability condition is examined, and [1], which is dedicated to studying expansions in simple theories (which generalize the stable ones).The well-known equivalence TP ⇔ TP 1 ∨ TP 2 due to Shelah [11] (where TP denotes the tree property while TP 1 and TP 2 denote the tree properties of the first and second kind, respectively), suggests two natural generalizations of simple theories, namely NTP 1 theories and NTP 2 theories (i.e., theories without TP 1 and TP 2 , respectively). So far, NTP 1 and NTP 2 theories have been studied much less extensively than the simple ones (i.e. theories without TP). However, recently some interesting results on these theories began to appear, notably [6] and [7]. In particular, natural examples of non-simple NTP 1 thoeries were provided in [6], namely: ω-free PAC fields, linear spaces with a generic bilinear form and a class of theories obtained by the "pfc" construction.The study of expansions in the NTP 2 context was undertaken in [2], where it was shown that the NTP 2 property is preserved under "dense and codense" unary predicate expansions of geometric structures where the unary predicate is assumed to define either an algebraically independent subset or an elementary substructure. In the present paper, we prove that the NTP 1 property is also preserved under such expansions. One of the main ingredients in our proof is the recently proved fact (due to Chernikov and Ramsey [6]) that the TP 1 property can, in any TP 1 theory, always be witnessed by some formula in a single free variable. We also prove (in Section 4) that an NTP 1 nonsimple geometric theory can be obtained from any Fraïssé limit which has a simple *

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