  We characterise when there exists a quasiisometric embedding between two
solvable Baumslag-Solitar groups. This extends the work of Farb and Mosher on
quasiisometries between the same groups. More generally, we characterise when
there can exist a quasiisometric embedding between two treebolic spaces. This
allows us to determine when two treebolic spaces are quasiisometric, confirming
a conjecture of Woess. The question of whether there exists a quasiisometric
embedding between two treebolic spaces turns out to be equivalent to the
question of whether there exists a bilipschitz embedding between two symbolic
Cantor sets, which in turn is equivalent to the question of whether there
exists a rough isometric embedding between two regular rooted trees. Hence we
answer all three of these questions simultaneously. It turns out that the
existence of such embeddings is completely determined by the boundedness of an
intriguing family of integer sequences.
