  In this paper we study the properties of the \emph{Triangular tree}, a
complete tree of rational pairs introduced in \cite{cas}, in analogy with the
main properties of the Farey tree (or Stern-Brocot tree). To our knowledge the
Triangular tree is the first generalisation of the Farey tree constructed using
the mediant operation. In particular we introduce a two-dimensional
representation for the pairs in the tree, a coding which describes how to reach
a pair by motions on the tree, and its description in terms of
$SL(3,\mathbb{Z})$ matrices. The tree and the properties we study are then used
to introduce rational approximations of non-rational pairs.
