  We study model theoretic tree properties ($\text{TP}, \text{TP}_1,
\text{TP}_2$) and their associated cardinal invariants ($\kappa_{\text{cdt}},
\kappa_{\text{sct}}, \kappa_{\text{inp}}$, respectively). In particular, we
obtain a quantitative refinement of Shelah's theorem ($\text{TP} \Rightarrow
\text{TP}_1 \lor \text{TP}_2$) for countable theories, show that $\text{TP}_1$
is always witnessed by a formula in a single variable (partially answering a
question of Shelah) and that weak $k-\text{TP}_1$ is equivalent to
$\text{TP}_1$ (answering a question of Kim and Kim). Besides, we give a
characterization of $\text{NSOP}_1$ via a version of independent amalgamation
of types and apply this criterion to verify that some examples in the
literature are indeed $\text{NSOP}_1$.
