L-functions of Carlitz modules, resultantal varieties and rooted binary trees

We continue study of some algebraic varieties (called resultantal varieties) started in a paper of A. Grishkov, D. Logachev Resultantal varieties related to zeroes of L-functions of Carlitz modules. These varieties are related with the Sylvester matrix for the resultant of two polynomials, from one side, and with the L-functions of twisted Carlitz modules, from another side. Surprisingly, these varieties are described in terms of finite weighted rooted binary trees. We give a (conjecturally) complete description of them, we find parametrizations of their irreducible components and their invariants: degrees, multiplicities, Jordan forms, Galois actions. Proof of the fact that this description is really complete is a subject of future research. Maybe a generalization of these results will give us a solution of the problem of boundedness of the analytic rank of twists of Carlitz modules.