I'm feeling really stupid but I can't understand why in this paper (page 3, right column) the number of nodes that are r hops away from the root of a tree with a branching factor equals to: $$(b+1)b^{r-1}$$ and the number of nodes not more than r hops away is: $$[(b+1)b^r-2]/(b-1)$$
I thought it should be b^r and (b^(r+1)-1)/(b-1) correspondingly.
Suppose a tree has constant valence or degree $b+1$ or branching factor $b$, by definition. The root has $b+1$ nodes adjacent to it in one hop. Each of those have a further $b$ nodes two hops from the root (hence branching factor), but they also are adjacent to the previous node which is the root. In general, any non-root node will have $b$ nodes which have distance from the root one greater, and $1$ node which is distance one less going back in the direction of the root. In other words, the root is special since all of its adjacent nodes are the same distance $1$ from itself, the root, but for all other nodes, their $b+1$ adjacent nodes split into $b$ nodes further from the root, and $1$ node closer to the root.
The two formulas regarding number of nodes at $r$ hops and at most $r$ hops are correct. Note that any $b$-ary tree can be isometrically embedded in the hyperbolic plane where each edge is given by a hyperbolic line segment and all have the same length which depends on $b$. The $b+1$ edges at any node are equally spaced angularly around that node with angle between adjacent edges $2\pi/(b+1)$.