Geodesic
Chebyshev polynomials, Zolotarev polynomials, and plane trees
Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 6, pp. 161-170.

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A polynomial with exactly two critical values is called a generalized Chebyshev polynomial (or Shabat polynomial). A polynomial with exactly three critical values is called a Zolotarev polynomial. Two Chebyshev polynomials $f$ and $g$ are called $\mathrm Z$-homotopic if there exists a family $p_\alpha$, $\alpha\in[0,1]$, where $p_0=f$, $p_1=g$, and $p_\alpha$ is a Zolotarev polynomial if $\alpha\in(0,1)$. As each Chebyshev polynomial defines a plane tree (and vice versa), $\mathrm Z$-homotopy can be defined for plane trees. In this work, we prove some necessary geometric conditions for the existence of $\mathrm Z$-homotopy of plane trees, describe $\mathrm Z$-homotopy for trees with five and six edges, and study one interesting example in the class of trees with seven edges. 
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     author = {Yu. Yu. Kochetkov},
     title = {Chebyshev polynomials, {Zolotarev} polynomials, and plane trees},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
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Yu. Yu. Kochetkov. Chebyshev polynomials, Zolotarev polynomials, and plane trees. Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 6, pp. 161-170. http://geodesic.mathdoc.fr/item/FPM_2013_18_6_a9/

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