In data mining and statistics, hierarchical clustering (also called hierarchical cluster analysis or HCA) is a method of cluster analysis that seeks to build a hierarchy of clusters. Strategies for hierarchical clustering generally fall into two categories:

• Agglomerative: This is a "bottom-up" approach: Each observation starts in its own cluster, and pairs of clusters are merged as one moves up the hierarchy.
• Divisive: This is a "top-down" approach: All observations start in one cluster, and splits are performed recursively as one moves down the hierarchy.

In general, the merges and splits are determined in a greedy manner. The results of hierarchical clustering are usually presented in a dendrogram.

The standard algorithm for hierarchical agglomerative clustering (HAC) has a time complexity of ${\displaystyle {\mathcal {O}}(n^{3})}$ and requires ${\displaystyle \Omega (n^{2})}$ memory, which makes it too slow for even medium data sets. However, for some special cases, optimal efficient agglomerative methods (of complexity ${\displaystyle {\mathcal {O}}(n^{2})}$) are known: SLINK for single-linkage and CLINK for complete-linkage clustering. With a heap, the runtime of the general case can be reduced to ${\displaystyle {\mathcal {O}}(n^{2}\log n)}$, an improvement on the aforementioned bound of ${\displaystyle {\mathcal {O}}(n^{3})}$, at the cost of further increasing the memory requirements. In many cases, the memory overheads of this approach are too large to make it practically usable.

Except for the special case of single-linkage, none of the algorithms (except exhaustive search in ${\displaystyle {\mathcal {O}}(2^{n})}$) can be guaranteed to find the optimum solution.

Divisive clustering with an exhaustive search is ${\displaystyle {\mathcal {O}}(2^{n})}$, but it is common to use faster heuristics to choose splits, such as k-means.

Hierarchical clustering has the distinct advantage that any valid measure of distance can be used. In fact, the observations themselves are not required: all that is used is a matrix of distances.