Hierarchical clustering studies a recursive partition of a dataset into clusters of successively smaller size, and is a fundamental problem in data analysis. In this work we study the cost function for hierarchical clustering introduced by Dasgupta, and present two polynomial-time approximation algorithms: Our first result is an O(1)-approximation algorithm for graphs of high conductance. Our simple construction bypasses complicated recursive routines of finding sparse cuts known in the literature. Our second and main result is an O(1)-approximation algorithm for a wide family of graphs that exhibit a well-defined structure of clusters. This result generalises the previous state-of-the-art, which holds only for graphs generated from stochastic models. The significance of our work is demonstrated by the empirical analysis on both synthetic and real-world datasets, on which our presented algorithm outperforms the previous state-of-the-art and several classical linkage heuristics.