Just a quick association to your tree structures and Fibonacci numbers.

The numbering scheme and hence the tree structure seems related to the concept of exponentiation by squaring. The power of a number (or matrix) can be recursively computed as follows.

• $A^{2n} = A^n * A^n $
• $A^{2n+1} = A * A^{2n} $

In this formula you recognize the left and right branches of your trees.

Computer scientists usually learn this when they see the Fibonacci sequence in their programming course. Straightforward recursion of the Fibonacci numbers leads to an exponential number of recursive calls, much worse that computing the numbers iteratively from the bottom. The Fibonacci numbers can be computed using matrix squaring using the matrices $\left(\begin{array}{cc}1 & 1 \\ 1 & 0 \end{array}\right)^n = \left(\begin{array}{cc}F_{n+1} & F_{n} \\ F_{n} & F_{n-1} \end{array}\right)$.

Just a quick association to your tree structures and Fibonacci numbers.

The numbering scheme and hence the tree structure seems related to the concept of exponentiation by squaring. The power of a number (or matrix) can be recursively computed as follows.

• $A^{2n} = A^n * A^n $
• $A^{2n+1} = A * A^{2n} $

In this formula you recognize the left and right branches of your trees.

Computer scientists usually learn this when they see the Fibonacci sequence in their programming course. Straightforward recursion of the Fibonacci numbers leads to an exponential number of recursive calls, much worse that computing the numbers iteratively from the bottom. The Fibonacci numbers can be computed using matrix squaring using the matrices $\left(\begin{array}{cc}1 & 1 \\ 1 & 0 \end{array}\right)^n = \left(\begin{array}{cc}F_{n+1} & F_{n} \\ F_{n} & F_{n-1} \end{array}\right)$.

The depth of the nodes is sequence A014701 in the On-Line Encyclopedia of Integer Sequences, "Number of multiplications to compute n-th power by the Chandah-sutra method".

Just a quick association to your tree structures and Fibonacci numbers.

The numbering scheme and hence the tree structure seems related to the concept of exponentiation by squaring. The power of a number (or matrix) can be recursively computed as follows.

• $A^{2n} = A^n * A^n $
• $A^{2n+1} = A * A^{2n} $

In this formula you recognize the left and right branches of your trees.

Computer scientists usually learn this when they see the Fibonacci sequence in their programming course. Straightforward recursion of the Fibonacci numbers leads to an exponential number of recursive calls, much worse that computing the numbers iteratively from the bottom. The Fibonacci numbers can be computed using matrix squaring using the matrices $\left(\begin{array}{cc}1 & 1 \\ 1 & 0 \end{array}\right)^n = \left(\begin{array}{cc}F_{n+1} & F_{n} \\ F_{n} & F_{n-1} \end{array}\right)$.

Just a quick association to your tree structures and Fibonacci numbers.

The numbering scheme and hence the tree structure seems related to the concept of exponentiation by squaring. The power of a number (or matrix) can be recursively computed as follows.

• $A^{2n} = A^n * A^n $
• $A^{2n+1} = A * A^{2n} $

In this formula you recognize the left and right branches of your trees.

Computer scientists usually learn this when they see the Fibonacci sequence in their programming course. Straightforward recursion of the Fibonacci numbers leads to an exponential number of recursive calls, much worse that computing the numbers iteratively from the bottom. The Fibonacci numbers can be computed using matrix squaring using the matrices $\left(\begin{array}{cc}1 & 1 \\ 1 & 0 \end{array}\right)^n = \left(\begin{array}{cc}F_{n+1} & F_{n} \\ F_{n} & F_{n-1} \end{array}\right)$.