  We study and classify faithfully balanced modules for the algebra of lower
triangular $n$ by $n$ matrices. The theory extends known results about tilting
modules, which are classified by binary trees, and counted with the Catalan
numbers. The number of faithfully balanced modules is a $2$-factorial number.
Among them are $n!$ modules with $n$ indecomposable summands, which can be
classified by interleaved binary trees or by increasing binary trees.
