Consider a tree in which each vertex has 5 children, and let us label the edges with $a$ or $b$ such that the edges from any parent to its children are always ordered $(a,b,a,b,b)$. See the figure for the case when there are 2 generations of children below the root. We allow $N$ generations in general.
Let us choose a vertex from the tree, at any level and consider its path back to the root. For instance, the left most grandchild has path $a\circ a$, the next is $a\circ b$ and so on.
Let us order the leaves of the tree from the left as $5(j-1)+k$ where $j$ is the index of the parent and $k=1,\dots,5$. In the example, the leaves will have labels 1 to 25.
Question 1
Given the label of a leaf, is there an expression for the path in terms of $a$ and $b$ in closed form. For instance, for some height $i$ and some index $j$ from $1$ to $5^{i-1}$ we might have $$ \text{leaf}_{ij} = a\circ b\circ \dots \circ a $$
If there is not, is there an accepted notation to describe this path?
Any help appreciated.
Thoughts
Paths are all the same length. Each position is either $a$ or $b$. If $j$ is $1,3,6,8,11,13, ...$ (congruent to $\{1, 3\}\mod 5$) then the element is $a$, otherwise it is $b$.
https://oeis.org/search?q=1%2C3%2C6%2C8%2C11%2C13&language=english&go=Search
Let $\ell$ be the index of the leaf, so $\ell$ is a number between $1$ and $5^N$. In your notation, $\ell=5(j-1)+k$.
In order to get the path sequence for the $\ell$, write $\ell-1$ in base five. The representation is a sequence of $N$ digits, each between $0$ and $4$. You may need to prepend some leading zeroes so the length is exactly $N$. To get the sequence of edge labels, substitute letters for the digits as follows: $$ 0\to a,\quad 1\to b,\quad 2\to a,\quad 3\to b,\quad 4\to b $$