In this document (pages 5-7), the Simplest Number Tree is used to explain how to assign a value to any arbitrary Blue-Red Hackenbush position. I'm having trouble following its approach.
I think I understand the "simplest number that fits" idea, at least in one direction; given a position value $x$, travel upward from vertex $x$ along the edges of the Simplest Number Tree, and $x =$ {the first encountered number less than (positioned to the left of) $x$ | the first encountered number greater than (positioned to the right of) $x$}. I'm not sure I understand the other direction, i.e., how to find $x$ given {$-\frac{3}{4} | \frac{7}{8}$}. And I certainly don't understand how the document is assigning a value to a pictorial Hackenbush position. It claims the process is inductive, but the example it uses is too trivial to demonstrate how it works.
Can someone clarify how use the tree to value an arbitrary Hackenbush position? A better description and perhaps a less trivial example may be helpful.
For finite numbers and dyadic fractions, if the left and right values straddle an integer take the smallest integer in absolute value, so your example would be $0$. If no integer fits you want the dyadic fraction with the smallest denominator that fits, so $\{\frac 5{16}|\frac 78\}=\frac 12$ or $\{3\frac 5{16}|3\frac{15}{32}\}=3\frac 38$