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Composition of plane trees. (English) Zbl 0914.05017

A bicolored plane tree \(T\) is a vertex bicolored tree with a prescribed cyclic order of edges adjacent to each vertex. A one-to-one correspondence exists between the set of bicolored plane trees and the set of equivalence classes of complex polynomials with at most two critical values called Shabat polynomials. (A polynomial \(P:{\mathcal C}\to{\mathcal C}\) is a Shabat polynomial if there exists a pair of complex numbers \(c_0\), \(c_1\) such that if \(P'(z)= 0\) then \(P(z)\) is equal to \(c_0\) or \(c_1\).) The tree corresponding to the polynomial \(P\) is denoted by \(A_P\). Given two Shabat polynomials \(P\) and \(Q\) for which the composition polynomial \(R(z)= P(Q(z))\) is a Shabat polynomial as well, one naturally expects the tree \(A_R\) to be constructible from the trees \(A_P\) and \(A_Q\). The authors present a construction of \(A_R\) from \(A_P\) and \(A_Q\), and include several examples of their construction.
If we denote the disjoint union of the counterclockwise rotations of the edges around the black vertices of a bicolored plane tree \(A_P\) by \(\pi_0\), and the corresponding “white” rotations by \(\pi_1\), the effective cartographic group of \(A_P\) is the permutation group \(G_P=\langle \pi_0,\pi_1\rangle\) acting on the set of edges of \(A_P\). A description of the effective cartographic group of the composition tree \(A_R\), \(R(z)= P(Q(z))\), in terms of the cartographic groups \(G_P\) and \(G_Q\) is presented.
The paper concludes with a series of remarks concerning possible further applications of the concepts introduced. We feel obliged, however, to point out that the paper contains essentially no proofs, and the material is presented in a rather informal way.

MSC:

05C05 Trees
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
12E10 Special polynomials in general fields
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