Ordering Trees 

There are many ways to order a tree.  If it contains n cases, or nodes, then there are 2n-1 different ways to order the cases at the base of the tree.  So, for example, with the Mammals data representing just 25 species, there are 224 = 16.77m different tree orders, for any one hierarchical cluster analysis.  This may come as a surprise, but you can check it out for yourself with a simple spreadsheet.  There are n-1 fusion steps producing a cluster, and at each step there are 2 possible ways to order the new cluster.  So the number of different tree orders is 2n-1.

Most clustering programs only give you one tree, if you're lucky.  Furthermore, it will be presented in an arbitrary order, determined by the order in which you listed your cases.  Change the order of the cases, and you will get a different tree.  It may be the same tree, but with the cases listed in a different order; or it may be a different tree, depending on whether there were any tied proximities.

In ClustanGraphics5, you get to select the tree you require.  For example, here is the standard tree produced for our single malt whisky case study.  We have selected four clusters by clicking the mouse at the 4-cluster partition.

Now point to the first cluster and click the right mouse button.  A pop-up menu appears which allows you to investigate the selected cluster in various ways; for example, by listing the cluster's membership, profiling cluster means for each of the variables, or displaying the cluster on a scatterplot.

Select "Invert" from the menu and click the mouse button.  The result is that the cluster of 5 malts Glenfiddich .. Edradour has been inverted, as shown left.  It's the same tree, but with a different case order down the left side.

After a few more clicks, you will easily obtain exactly the tree order you need.  For example, the tree below lists the whiskies from the heaviest, dry, peaty island cluster (Laphroaig) to the sherried, sweet, Speyside cluster (Macallan).  The two extremes of Scotch malt whisky, some would argue.

Now that you can appreciate the many ways of ordering any tree from a hierarchical cluster analysis, perhaps you would like to find the optimum order.  This is explained in reorder tree where we maximize the rank correlation of the proximity matrix for any given tree.

It's a class act!


Clustan - A Class Act © 1998 Clustan Ltd.