  In this paper we present new proofs of the non-embeddability of countably
branching trees into Banach spaces satisfying property $(\beta_p)$ and of
countably branching diamonds into Banach spaces which are $p$-AMUC for $p > 1$.
These proofs are entirely metric in nature and are inspired by previous work of
Ji\v{r}\'i Matou\v{s}ek. In addition, using this metric method, we succeed in
extending these results to metric spaces satisfying certain curvature-like
inequalities. Finally, we extend an embedding result of Tessera to give lower
bounds on the compression for a class of Lipschitz embeddings of the countably
branching trees into Banach spaces containing $\ell_p$-asymptotic models for $p
\geq 1$.
