The algorithm for building wheel trees is given in the figure below.
The procedure accepts phylogenetic trees containing the same set of taxonomic units and first builds a consensus tree with a given threshold. It can also accept weight values that reflect observed frequencies or probabilities of the trees.
Then, for each multifurcating node (v) on the consensus tree, the best circular ordering of the branches adjacent to v (v-branches) is calculated through the following steps.
After repeating the above for all multifurcating nodes, finally visualize the consensus tree by using the TSP circular orderings for branches around each multifurcation (wheel node).
Wheel trees produced by the procedure outlined above not only embodies the Key Concept. Mathematically, it is the centroid representation, which is the most balanced representation of the tree distributions. An intuitive explanation is as follows:
Imagine a heap of stones, each of which has a drawing of a phylogenetic tree and whose mass is proportional to the probability of the tree.
If we place the stones so that similar trees are close to each other, a map of a tree distribution is created.
If this distribution is unimodal and unbiased, the heaviest stone or the best tree is anticipated to be at the "center" of the distribution and represent it well, although this assumption cannot hold true for estimation problems in ultra-high dimensional spaces.
Instead, the wheel tree representation finds the centroid (or the center of gravity) of the trees, by placing them on a light plate on which each position corresponds to a wheel tree.