[seqfan] Asymptotic behaviour of A034824 and friends

[Rob Arthan]

I would like to estimate how many trees there are with height at most 8 and with at most 2^64 nodes.
I.e., I want to estimate the sum of the first 2^64 terms of A034824. Does anyone know the
asymptotic behaviour of this sequence? Or even better, the general case for height at most H
and at most N nodes, with H fixed and with N tending to infinity? 

Regards,

Rob.