## Theory

**Definition.** *A hierarchy is a single-rooted tree.*

Let \(T=(V,E)\) be a connected, acyclic directed graph that expresses a hierarchical structure. The set of vertices \(V\) represents the entities that are being organized, and the set of edges \(E\) maps from a given entity to its respective superior.

A vertex is said to have degree \(k\) if it has exactly \(k\) direct subordinates (incoming edges). Vertices with degree 0 are called leaves and vertices with degree \(>0\) are called nodes.

**Definition.** *The average degree of a hierarchy is the average degree of its nodes.*

If \(T\) is not empty, then there exists exactly one distinguished vertex \(v_1\in{}V\) that has no superior. All other vertices have exactly one superior.

For every \(v\in{}V\) there exists a unique path that leads from \(v\) to \(v_1\). The length of that path is referred to as depth of \(v\). The depth of \(v_1\) is 0.

**Definition.** *The average depth of a hierarchy is the average depth of its leaves.*

Let \(\mu\) be the average degree of \(T\), and let \(l\) be the average depth of \(T\). Then there exists a perfectly regular tree \(T'\) in which *all* nodes have degree \(\mu\) and *all* leaves are at depth~\(l\):

Such a tree has \(\mu^k\) vertices at any given depth \(k\), which means that the total number of vertices is \[ n \;=\; \sum\limits_{k=0}^{l}\,\mu^k \;=\; \frac{1-\mu^{l+1}}{1-\mu} . \]

**Corollary.** *In every perfectly regular hierarchy, the relationship between the average degree \(\mu\), the average depth \(l\), and the number of vertices \(n\) is \[n\,\mu^{l+1}-\mu^{l}-n+1\;=\;0 .\]*

**Corollary.** *The proportion of nodes in every perfectly regular hierarchy is \[\frac{1-\mu^{l}}{1-\mu^{l+1}} .\]*

## Application

Consider any given company hierarchy of \(n=10\,000\) employees. Nodes represent managers and leaves represent workers. Workers have an average chain of \(l=4\) superiors up to (and including) the big boss. If that hierarchy is somewhat regular, then approximately 10% of the workforce — \(1\,000\) employees — would be required for purposes of management. The average team in that company would have approximately \(\mu=9,7\) members. If the company match these predictions, then its hierarchy would have to be somewhat irregular.