This is called a Galton-Watson tree. From one ancestor (the root of the tree), we have a random number
of children, each of which itself has random numbers of children, etc., where the numbers of children are all independent of each other. This is a genealogic tree where the numbers of children is random: k children (possibly 0) with probability
. A basic question is: does the family eventually get extinct? i.e. is the tree finite?
In this case,
would be the set of trees (or a larger set), or an equivalent representation of trees. A convenient choice is to let
, the set of all finite sequences of integers, and
, the set of positive-integer sequences indexed by elements of
. Intuitively, an individual corresponds to a sequence
if it is obtained from the root as the
-th child of the
-th child of .... of the
-th child of the root. And the number of children of this individual is encoded in the index
of the tree
is not connected to the root, thus there are more codings than trees).