"Counting" the Integers in the Abstract Predecessor Tree via Path Densities

-- counting infinite sets is tough and hazardous
-- avoid the traps by adding densities (or relative densities)
-- taking the density of the extensions as 1,
           -the immediate leaf set is 1/3 the density of the extensions
           -2 first level children are 1/3 leaf nodes, whence 2/9 of density are leaves
           -4 second level children are 1/3 leaf nodes, whence 4/27 of density are leaves
           -sum(2ⁱ/3^⁽ⁱ⁺¹⁾,i=1..infinity) is 1
-- whence all the extension elements ultimately terminate in leaves (are in l.d.a.s; are in the abstract predecessor tree)
-- Similarly, all the odd positive integers are in the abstract predecessor tree.
-- a gross (i.e. monolithic) summation over the whole predecessor tree