Hierarchical clustering is a popular unsupervised data analysis method. For many real-world applications, we would like to exploit prior information about the data that imposes constraints on the clustering hierarchy, and is not captured by the set of features available to the algorithm. This gives rise to the problem of "hierarchical clustering with structural constraints". Structural constraints pose major challenges for bottom-up approaches like average/single linkage and even though they can be naturally incorporated into top-down divisive algorithms, no formal guarantees exist on the quality of their output. In this paper, we provide provable approximation guarantees for two simple top-down algorithms, using a recently introduced optimization viewpoint of hierarchical clustering with pairwise similarity information [Dasgupta, 2016]. We show how to find good solutions even in the presence of conflicting prior information, by formulating a constraint-based regularization of the objective. We further explore a variation of this objective for dissimilarity information [Cohen-Addad et al., 2018] and improve upon current techniques. Finally, we demonstrate our approach on a real dataset for the taxonomy application.