A187131 Numerator of probability that the height of a rooted random binary tree is n.

1, 1, 9, 1521, 71622369, 233297499911422401, 3390052406222940758260506721830900609, 934785860242188709610961043825803400592180434378516146129897302939414193921

Offset: 0

Author

Henry Bottomley, Mar 05 2011

Keywords

• nonn

Comments

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;

* Sum_n p(n) = 1;

* Sum_n n*p(n) is infinite;

* p(n) = a(n) / 2^(2^(n+1)-1).

Examples

			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.
		

Crossrefs

Denominator is A058891 offset

Formula

a(n) = A076628(n)^2.