## One more Nested Intervals encoding

### June 20, 2008

Dan Hazel developed one more version of continued fraction tree encoding. It is equivalent to matrix based approach with atomic matrices of the kind

`[n + 1 n]`

[ ]

[ 1 1]

but should be appealing to the readers who find continued fractions more intuitive compared to matrices.

Here is matrix perspective to his work. The atomic matrix in the foundation of Dan’s approach is

`[n + 1 n]`

[ ]

[ 1 1]

It has determinant equal to one, therefore general path determinants are not alternating, and nested interval orientation does not change when we ascend to the next tree level. Each Dan’s atomic matrix is a product of the familiar atomic matrix

`[n 1]`

[ ]

[1 1]

with an auxiliary matrix

`[1 1]`

[ ]

[1 0]

This is why Dan’s continued fractions contain interleaving sequences of 1s.