A002449 as Solution to System of Equations?

[Paul D. Hanna]

Seqfans,
      Help is needed to simplify the following problem.
 
Consider the infinite system of simultaneous equations:
A = 1 + xAB
B = 1 + xBCD
C = 1 + xCDEF
D = 1 + xDEFGH
E = 1 + xEFGHIJ
F = 1 + xFGHIJKL
... 
What is the unique solution to the variables A,B,C,... as functions in x?

  
Is there a way to further simplify the above system of equations 
to minimize the computational effort to calculate the 
sequences for A,B,C,...?   Ex., B = (A-1)/(xA), etc. 
   
I computed the initial coefficients of A to be: 
[1,1,2,6,26,166,1626,25510,664666,29559718,...] 
but time did not permit me to get more terms since 
the above system of equations (in this raw form) 
requires (n+1) iterations of the first 2^n equations 
involving 2^(n+1) power series A,B,C,... O(x^(2^n)) 
just for accuracy on the coefficient of x^n in A!  
  
I believe that A = A(x) is the g.f. of A002449 
(number of different types of binary trees of height n): 
[1,1,2,6,26,166,1626,25510,664666,29559718,2290267226,...] 
due to the coincidence of the initial terms, as well as 
the binary nature of the system of equations. 
  
Could someone calculate more coefficients of A 
to see if they continue to agree with A002449? 
  
Thanks, 
       Paul 
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