Chebyshev polynomials, Zolotarev polynomials, and plane trees

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.