Resolving the double-root conjecture in Ballot–Kimberling–Moses (2017), Section 5, Table 5, p. 23

We prove the conjectured recurrence for q(x)=x^2-2x+1 in Ballot–Kimberling–Moses (2017), Section 5, Table 5, page 23. The polynomial-form tree has G_n=1+binomial(n,2) for every n>=0, so its recurrence with coefficients (3,-3,1) holds from n=3. Value-and-derivative coordinates give an exact shortest-path formula. We also prove G_n=n for n>=1 modulo x^2-1, correcting G_3 from 4 to 3, and G_n=6n-20 for n>=6 modulo x^2-x+1, with recurrence starting at n=8. For x^m-1 we give exact distances and the count sum of binomial(n-1,r) for 0<=r=1.