Trees with non log-concave independent set sequences

We construct a family of trees with independence numbers going to infinity for which the log-concavity relation for the independent set sequence of a tree $T$ in the family fails at around $\alpha (T)(1-1\mathrm{/}(16\log \alpha (T)))$. Here $\alpha (T)$ is the independence number of $T$. This resolves a conjecture of Kadrawi and Levit.