Any metric $d: X \times X \to \mathbb{R}$ on a set $X$ must satisfy the triangle inequality:

\begin{equation*} (\forall x,y,z) \quad d(x,z) \leq d(x,y) + d(y,z) \end{equation*} An ultrametric must additionally satisfy a stronger version of the triangle inequality:

\begin{equation*} (\forall x,y,z) \quad d(x,z) \leq \max\{ d(x,y), d(y,z) \} \end{equation*} Here is an example of an ultrametric on a space with 5 points, labelled $a,b,c,d,e$ :

In the table above, an entry $n$ in the row for element $x$ and the column for element $y$ indicates that $d(x,y)=n$ , where $d$ is the ultrametric. By symmetry of the ultrametric ($d(x,y)=d(y,x)$ ), the above table yields all values of $d(x,y)$ for all $x,y \in \{a,b,c,d,e\}$ .

The ultrametric condition is equivalent to the ultrametric three point condition:

\begin{equation*} (\forall x,y,z) \quad x,y,z \textrm{ can be renamed such that } d(x,z) \leq d(x,y) = d(y,z) \end{equation*} Ultrametrics can be used to model bifurcating hierarchical systems. The distance between nodes in a weight-balanced binary tree is an ultrametric. Similarly, an ultrametric can be modelled by a weight-balanced binary tree, although the choice of tree is not necessarily unique. Tree models of ultrametrics are sometimes called ultrametric trees.

The metrics induced by non-Archimedean valuations are ultrametrics.