A BC-tree (block-cutpoint-tree) is a tree (with at least two vertices) where the distance between any two leaves is even. A BC-subtree is a subtree of a connected graph that is also a BC-tree. In this paper, we first consider subtrees containing a specific vertex with conditions on the parity of the distances from this vertex to the leaves and the enumeration of such subtrees of unicyclic and edge-disjoint bicyclic graphs. Using these results and through generating functions of BC-subtrees, we present graph-theoretical based algorithms for enumerating BC-subtrees, BC-subtrees containing a given vertex, and BC-subtrees containing two distinct vertices of unicyclic and edge-disjoint bicyclic graphs. These results also provide a novel perspective on exploring the structural properties of molecules.