On model-theoretic tree properties

We study model theoretic tree properties (TP, TP1, TP2) and their associated cardinal invariants (k(cdt), k(sct), k(inp), respectively). In particular, we obtain a quantitative refinement of Shelah's theorem (TP double right arrow TP1 V TP2) for countable theories, show that TP1 is always witnessed by a formula in a single variable (partially answering a question of Shelah) and that weak k - TP1 is equivalent to TP1 (answering a question of Kim and Kim). Besides, we give a characterization of NSOP1 via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are indeed NSOP1.