Sequence A112088 Motivation and Example?

[Rainer Rosenthal]

To SeqFan and to author Simon Strandgaard:

I am very much interested in this sequence:
http://www.research.att.com/~njas/sequences/A112088.

I don't understand how Simon generated this
sequence. There is some program code but there 
should be one example at least. (I thought this
to be mandatory if you submit a sequence.)

I am quite sure that this is somehow related to
binary trees and balancing. And there is a remark
of Knuth in the end of his exercise 5.1.1-5, which
in my opinion is related to this sequence. I don't
want to discuss my own Josephus based generation
of this sequence but ask everybody here at SeqFan
to please help me: What is that sequence all about?
May be Robert G. Wilson can tell me, because he is
named as the author of the Mma program and the
extension of this sequence. BTW I'm not used to Mma:
seeing Array[f,40] I expect 40 values f(1)...f(40).
But I see only 39 members listed.

TIA and best regards,
Rainer Rosenthal
r.rosenthal at web.de