Unbounded log-concavity breaks in independence polynomials of spherically symmetric trees

Using a dioid algebraic structure, we show that there exist spherically symmetric trees $T(2^m{1}^n)$ whose independence polynomials exhibit multiple breaks in log-concavity, a result established by estimating the asymptotic growth of the coefficients of these polynomials. Provided the parameter $n$ is a sufficiently large odd integer, the number of breaks is bounded below by the Jacobsthal numbers. This result affirmatively answers a question raised by D. Galvin [D. Galvin, Trees with non log-concave independent set sequences, arXiv 2502.10654.v2].