Lengths of Infinite Farey Tree continued fractions

One half of the Infinite Farey tree, refer to http://www.tinyurl.com/4zq4q, then enter sequence
A088696:

1/2

1/3 2/3

1/4 2/5 3/5 3/4

1/5 2/7 3/8 3/7 4/7 5/8 5/7 4/5

...

and their corresponding continued fraction representations are:

[2]

[3] [1,2]

[4] [2,2] [1,1,2] [1,3]

[5] [3,2] [2,1,2] [2,3] [1,1,3] [1,1,1,2] [1,2,2] [1,4]

...

with the number of terms in each continued fraction representation generating A088696:

1

1 2

1 2 3 2

1 2 3 2 3 4 3 2

where the latter as a sequence may be considered the infinite linear string of the natural numbers based on the Gray code reflection principle. Refer to the entry, "Natural numbers, Gray code string"