Reading Dendrograms And Choosing A Cut

Reading a Dendrogram

A dendrogram displays the nested groups created by a hierarchical clustering procedure. Its leaves represent the individual observations being clustered; its branches show how groups combine; and its vertical axis records the height, or dissimilarity value, at which each combination occurs. Reading the tree carefully lets you recover cluster memberships at different stages without rerunning the clustering.

The key is to follow the branches from the leaves upward. A horizontal cut separates the observations into the groups that have joined below the cut but have not yet joined one another above it. Moving the cut higher generally produces fewer groups in a monotone dendrogram. The decision about which cut is useful requires interpretation: the tree describes nested groupings, but does not by itself establish a uniquely correct number of clusters. This skill is aligned with the SOA Exam PA (Predictive Analytics) syllabus.

In a binary agglomerative tree with \(n\) leaves, each merge reduces the number of separate groups by one. A complete tree therefore contains \(n-1\) merges: it begins with \(n\) singleton groups and ends with one group containing all observations. Each merge must combine groups that already exist. For example, after forming \(\{A,B\}\) and \(\{C,D\}\), the next merge could join those two groups to make \(\{A,B,C,D\}\); it cannot add \(C\) to \(\{A,B\}\) while also leaving \(C\) in \(\{C,D\}\).

Leaf order along the bottom is not usually meaningful by itself. The branches convey the nested memberships; branches can often be flipped left-to-right without changing the clustering. Nor does a branch's visual width or the horizontal distance between leaves generally represent the merge height. Read the value from the vertical axis at the merge.

How to Determine Memberships at a Cut

For a monotone dendrogram, a merge at height less than or equal to \(h\) has occurred by a cut at \(h\); a merge above \(h\) has not. At a height exactly equal to a merge, this tutorial uses the convention that the merge is included. In practice, a cut between two distinct merge heights avoids ambiguity about equality. To identify the groups, start at each leaf and trace its branch upward only as far as the cut. Leaves connected below the cut form one cluster.

The count has a useful check. If the tree is monotone and \(m\) merges have occurred by the cut, then the number of remaining clusters is \(n-m\). This follows because there are initially \(n\) singleton groups, and each included merge replaces two groups with one, reducing the count by exactly one. The count check is helpful, but it does not replace tracing the branches: it cannot tell you which observations belong together.

Worked Cuts

Choosing and Explaining a Cut

A cut is a way to turn a nested tree into one flat grouping. The vertical axis lets you state where groups combine, and the cut specifies which combinations to retain. A lower cut preserves more, smaller groups; a higher cut combines them into fewer, larger groups. To explain a chosen cut clearly, report the height or interval, the memberships, and the resulting count. If the purpose is to describe groups, also say which observations are kept together or remain separate.

There is not necessarily one best cut for every purpose. A cut may be selected to create a manageable number of groups, to preserve a visually distinct separation in the tree, or to support a later analysis. These are different considerations. The tree alone does not establish that a group is useful, predictive, or meaningful in context. A cluster is a grouping based on the inputs and dissimilarities used to construct the tree, not a discovered outcome label.

A horizontal line between two consecutive merge heights can be especially convenient. Every cut in that interval includes the same merges and therefore produces the same memberships. If two or more merges share a height, a cut exactly at that height can include several joins at once under the convention used here; state the convention or choose a cut just above or below the tied height when the distinction matters.

The cluster-count calculation also has a limitation. The simple rule “leaves minus merges at or below the cut” assumes a monotone dendrogram, where merge heights do not reverse the vertical ordering of earlier and later joins. If a displayed algorithm produces an inversion, do not mechanically count historical merges by their listed heights. Read the branches where the horizontal line intersects them and inspect how the tree represents the groups. The examples here use monotone merge heights.

Common Mistakes and a Reliable Check

A frequent error is to count leaves below the cut instead of groups. Five leaves do not imply five clusters if some of their branches have already joined. The reverse error is to count branch junctions above the cut as if they had already happened. A merge above the line has not joined its two groups at that cut.

For example, in the five-leaf tree above, at height \(3.0\) the merge at \(2.5\) has happened, so \(\{A,B\}\) and \(\{C,D\}\) form one group. A wrong calculation might leave those as two groups because it overlooks the \(2.5\) merge, and report three groups: \(\{A,B\}\), \(\{C,D\}\), and \(\{E\}\). The correction is to include every merge below the cut: the actual groups are \(\{A,B,C,D\}\) and \(\{E\}\), for two clusters.

Another trap is to infer membership from nearby leaf labels or to treat a merge height as a count. Height \(2.5\) is a dissimilarity-axis value, not “two and a half clusters,” and two leaves that appear next to one another on the page need not be the first pair to merge. Trace the branches and check that every leaf appears exactly once.