The binomial tree of order 0 is a single root vertex and order n>=1 is an order n-1 with another order n-1 joined as a subtree of the root.
Going by induction with this construction shows the number of sets where the root is in the set is with(n) =
A052129(n), and the number where the root is not in the set is without(n) =
A088679(n+1).
Among the total a(n), the respective proportions are without(n)/a(n) = (n+1)/(n+2) and with(n)/a(n) = 1/(n+2).
Also, a(n-1) is the number of maximum independent sets in binomial tree order n.
Tree n can be constructed from tree n-1 by adding a new child under each vertex. Each independent set in tree n-1 corresponds one-to-one with a maximum independent set in tree n by putting each new child in or out of the set opposite to its parent.
