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If each node of a rooted random binary tree has probability 1/2 of producing two branches, and p(n) is the probability that the height of the tree is n, then p(n) has the following properties:
* p(n) = 2*b(n+1)^2 with b(n) defined as in A076628;
* p(n+1) = p(n)*(1 - sqrt(p(n)/2))^2 starting from p(0)=1/2;
For n=0 the root node may have no branches giving the tree height 0, so p(0)=1/2 and a(0)=1; p(1) = 1/2*1/4 = 1/8 so a(1)=1; p(2) = 1/4*1/4 + 1/8*1/16 = 9/128 so a(2)=9; p(3) = 5/32*1/4 + 7/64*1/16 + 1/32*1/64 + 1/128*1/256 = 1521/32768 so a(3)=1521.