Numerator/Denominator Tree: Continued-Fraction Recursion

[Leroy Quet]

Posted this to sci.math:

Consider the tree of positive integers defined as follows:

Only term in first row is 1.

Term-(2k-1) in row-(m+1) is numerator of the (reduced) value of the 
continued fraction,
[a(1);a(2),a(3),...a(m)],

and term-(2k) in row-(m+1) is the denominator of this CF, where

a(j) = term-n in row-j which is connected to term-ceiling(n/2) in 
row-(j-1).
And a(m) is term that term-(2k-1) and term-(2k) in row-(m+1) are 
connected to (ie. term-k in row-m).

If I have calculated correctly, the tree begins as:

                    1
              /            \
           1                   1
       /       \            /      \
      2         1          2         1
    /   \      /  \      /  \       /  \
   5     3    3    2    5    3      3   2
  / \   / \  / \  / \  / \   / \   / \ / \
27  16 17 10 11 7 8 3 27 16 17 10 11 7 8  3

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 What is the sequence that gives:
a(j) = total number of times that j appears in the tree?

-

The tree's values can be arranged into concentric circles, so that the 
m_th row is written in the m_th concentric circle from the center, and 
the terms are aligned in the circles as they are in the tree (adjacent to 
same values).

 If these values are taken, for example, mod n, then which n's form a 
beautiful/interesting design if each value is assigned one of n colors?

Thanks,
Leroy Quet