  A polynomial with exactly two critical values is called a generalized
Chebyshev polynomial. A polynomial with exactly three critical values is called
a Zolotarev polynomial. Two Chebyshev polynomials $f$ and $g$ are called
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),
Z-homotopy can be defined for plane trees. In this work we prove some necessary
geometric conditions for plane trees Z-homotopy, describe Z-homotopy for trees
with 5 and 6 edges and study one interesting example in the class of trees with
7 edges.
