Tree with branching factor 1
The formula you state follows from summing a geometric series:
$$
1 + b + b^2 + \cdots + b^d = \frac{b^{d+1}-1}{b-1}.
$$
When $b=1$, the series sums to $d+1$, which is also the limit of the expression above as $b\to1$.

As you mention, the formula is only meaningful when $b \neq 1$, though the correct value can still be recovered by taking a limit. This is perhaps the simplest example of a curious phenomenon in combinatorics, that of the "field with one element". Consider the following two quantities:


The number of subsets of size $k$ of a set of size $n$. These are counted by binomial coefficients $\binom{n}{k}$.
The number of $k$-dimensional subspaces of an $n$-dimensional vector space over the finite field with $q$ elements. These are counted by Gaussian binomial coefficients $\binom{n}{k}_q$, which are rational functions (ratios of two polynomials) in $q$.


The curious thing is that if we substitute $q=1$ in the expression for $\binom{n}{k}_q$, we get back $\binom{n}{k}$ in the limit. (Just as in your case, when $q=1$ the denominator vanishes.) So a set is similar to a vector space over "the field with one element".

More generally, there are other formulas involving "$q$-analogs" (sets replaced with vector spaces over the field with $q$ elements) which degenerate to the correct values when taking the limit $q\to1$. There are some explanations for this phenomenon but it is still somewhat mysterious.