## "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(2^{i}/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

next slide
return to slide index