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In a weighted undirected graph, a spanning/Steiner shallow-light tree (SLT) simultaneously approximates: 1) shortest distances from a root to the other vertices and 2) the minimum tree weight. The Steiner SLT has been proved to be exponentially lighter than the spanning one. In this paper, we propose a novel Steiner SLT construction method called Steiner SLT (SALT), which is efficient and has the tightest bound over all the state-of-the-art general-graph SLT algorithms. Applying SALT to Manhattan space offers a smooth tradeoff between rectilinear Steiner minimum tree and rectilinear Steiner minimum arborescence for VLSI routing. The adaption also reduces the time complexity from ${O}$ (${n} ^{2}$) to ${O}$ (${n}$ log ${n}$). Besides, several effective post-processing methods, including safe refinement and shallowness-constrained edge substitution, are proposed to further improve the result. The experimental results show that SALT can achieve not only short path lengths and wirelength but also small delay, compared to both classical and recent routing tree construction methods. [ABSTRACT FROM AUTHOR]