Denote by \(\hat K\) the field of formal Laurent series over a finite field \(\mathbb F_q\). Consider an element \(f\in \hat K\) which is quadratic irrational element over \(\mathbb F_q(Y)\), and let \(P\) be any irreducible polynomial in \(\hat K\). The authors study the asymptotic properties of the degrees of the coefficients of the continued fraction expansion of the sequence of quadratic irrationals \(P^n f\) (\(n=1,2,\ldots\)). They prove that they have one such degree very large with respect to the other ones. The authors employ discrete geodesic flows on the Bruhat-Tits building of \((\mathrm{PGL}_2,\hat K\)) and the Hecke trees.