In this paper, we propose a general scheme, called Algorithm $\mathsf{STlC}$, to compute spanning-tree-like data structures on arbitrary networks. $\mathsf{STlC}$ is self-stabilizing and silent and, despite its generality, is also efficient. It is written in the locally shared memory model with composite atomicity assuming the distributed unfair daemon, the weakest scheduling assumption of the model. Its stabilization time is in $n_{\texttt{maxCC}}$ rounds, where $n_{\texttt{maxCC}}$ is the maximum number of processes in a connected component. We also exhibit polynomial upper bounds on its stabilization time in steps and process moves holding for large classes of instantiations of Algorithm $\mathsf{STlC}$. We illustrate the versatility of our approach by proposing several such instantiations that efficiently solve classical problems such as leader election, as well as, unconstrained and shortest-path spanning tree constructions.