  It is proved that if $G$ is a graph containing a spanning tree with at most
three leaves, then the chromatic polynomial of $G$ has no roots in the interval
$(1,t_1]$, where $t_1 \approx 1.2904$ is the smallest real root of the
polynomial $(t-2)^6 +4(t-1)^2(t-2)^3 -(t-1)^4$. We also construct a family of
graphs containing such spanning trees with chromatic roots converging to $t_1$
from above. We employ the Whitney $2$-switch operation to manage the analysis
of an infinite class of chromatic polynomials.
