Given a squarefree monomial ideal $I$ of a polynomial ring $Q$, we show that if the minimal free resolution $\mathbb{F}$ of $Q/I$ admits the structure of a differential graded (dg) algebra, then so does any ``pruning" of $\mathbb{F}$. In the language of combinatorics, this says that if $Q/\mathcal{F}(Δ)$, the quotient of the ambient polynomial ring by the facet ideal $\mathcal{F}(Δ)$ of a simplicial complex $Δ$, is minimally resolved by a dg algebra, then so is the quotient by the facet ideal of each facet-induced subcomplex of $Δ$ (over the smaller polynomial ring). Along with techniques from discrete Morse theory and homological algebra, this allows us to give complete classifications of the trees and cycles $G$ with $Q/I(G)$ minimally resolved by a dg algebra in terms of the length of the longest path in $G$, where $I(G)$ is the edge ideal of $G$.