I learned that the reason $a^0$ is defined as $1$ is for consistency with the rule $a^ba^c=a^{b+c}$. However, the fact that $a^0$ is $1$ is used in contexts outside of the algebraic manipulation seen in the prior sentence. It is unclear to me what the relationship is between those contexts and the definition of $a^0$.
For instance, we can expand an integer $k$ in some base $b$ as $k = \sum_{i=0} c_ib^i.$ This expansion is predicated on the fact that $b^0=1$.
Another context comes from looking at the maximum number of nodes at level L in a binary tree: $2^L$. If we let $L=0$ for the root node we get the correct maximum number of root nodes: 1.
Are these examples somehow fundamentally connected to the algebraic motivation for defining $a^0$ as $1$? If so, what is that connection?
