Hierarchical clustering — the family tree of your data

The final module drops the labels. Given only points, find the groups — but this time without k-means' awkward demand that you announce the number of clusters up front. Agglomerative clustering sidesteps it with a simple bottom-up ritual: start with every point as its own cluster, then repeatedly merge the two closest clusters, recording the distance at which each merge happened, until one cluster remains.

The record of merges is the output: the dendrogram, a tree whose joints sit at their merge distances. Low joints are easy, obvious unions; a tall joint is the algorithm holding its nose while gluing two genuinely different groups. To get concrete clusters, slice the tree at any height — and here's the elegance: one tree contains the clustering for every possible K at once. Slide the cut; the answer reorganizes instantly, no refitting.

One definitional choice hides real consequences: what does “distance between clusters” mean once clusters hold many points? Single linkage (closest pair) chains through touching neighbors and grows straggly clusters; complete linkage (farthest pair) demands total cohesion and grows compact ones; averagesplits the difference. Same data, different trees — try all three below.