We investigate the existence of closed copies of the discrete space $\mathbb{N}$ of natural numbers in powers of the real line, in particular its $\omega_1$-power, that are not $C^\star$-embedded, or that are $C^\star$-embedded but not $C$-embedded. In the case of non-$C^\star$-embedding we find a whole family of new examples, based on Aronszajn trees and lines, and a combinatorial translation of the existence of such copies. In the case of $C^\star$- but not $C$-embedding we complement an earlier consistency result but showing in consistent with any desired cardinal arithmetic that $\mathbb{R}^{\omega_1}$ contains a closed copy of $\mathbb{N}$ that is $C^\star$- but not $C$-embedded.