We identify a structural pattern in the construction of known infinite families of trees whose independence polynomials are not log-concave. Using this pattern and properties of polynomial ring ideals, we derive linear recurrences for these polynomials. As a consequence, we prove that the set of non-isolated limit points of their zeros lies on the circle $|z+1/3|=1/3$ in the complex plane. Building on these recurrences, we also exhibit infinite families of trees whose independence polynomials break log-concavity at one, two, and three consecutive indices, as well as finite families that break log-concavity at four and five consecutive indices. Our approach suggests that arbitrarily many consecutive breaks may be achievable, offering further insight into a question posed by Galvin [D. Galvin, Trees with non log-concave independent set sequences, arXiv:2502.10654v1, 2025].