I am taking the following as the definition of a $\Delta$-complex.
(i) one starts with an indexing set $I_n$ for each $n \in \mathbb{Z}_{\ge 0}$.
(ii) for each $\alpha \in I_n$, one takes a copy $\sigma_\alpha^n$ of the standard $n$-simplex.
(iii) one forms the disjoint union of all of these simplices, for all $n\geq 0$.
(iv) now require that for each $(n-1)$-dimensional face of $\sigma_\alpha^n$, there is an associated $\sigma_\beta^{n-1}$ for some $\beta \in I_{n-1}$.
(v) now form the quotient space by identifying each $(n-1)$-dimensional face of each $\sigma_\alpha^n$ with $\sigma_\beta^{n-1}$ using the canonical homeomorphism. In particular, these homeomorphisms preserve the ordering of vertices.
The sources I've consulted don't make it clear what a subcomplex is. Can someone give me a rigorous definition?