Say I only wanted to enumerate the rational numbers between 0 and $a$. Is there a way to "narrow" a Stern-Brocot tree to provide this? I tried keeping my left bound at "$\frac{0}{1}$" and setting my right-bound to "$a$" (where $a$ obviously is a rational) but if I set $a$ to $\frac{11}{10}$, then my first mediant is "1" and my tree is horribly skewed. Setting the left bound to be "$\frac{0}{10}$" alleviates this, but introduces a new problem in that all of my fractions are now non-reduced (modified tree enumerates $\frac{55}{110}$ instead of $\frac{1}{2}$).
Is there a "better" way of doing this?
There is just one Stern-Brocot tree, and if you fiddle with the definition you'll get something else, which moreover does not have the properties of the Stern-Brocot tree. The intervals of the rational numbers that have a Stern-Brocot-like tree above them with "all" the properties you want are probably precisely the ones that are beneath a given rational number in the Stern-Brocot tree itself. For instance everything below $\frac7{11}$ is $(\frac58,\frac23)$ so that interval is OK.