  The independence polynomial of a graph $G$ is \[I(G,x)=\sum\limits_{k\ge
0}i_k(G)x^k,\] where $i_k(G)$ denotes the number of independent sets of $G$ of
size $k$ (note that $i_0(G)=1$). In this paper we show a new method to prove
real-rootedness of the independence polynomials of certain families of trees.
  In particular we will give a new proof of the real-rootedness of the
independence polynomials of centipedes (Zhu's theorem), caterpillars (Wang and
Zhu's theorem), and we will prove a conjecture of Galvin and Hilyard about the
real-rootedness of the independence polynomial of the so-called Fibonacci
trees.
