This note presents two results on real zeros of chromatic polynomials. The first result states that if $G$ is a graph containing a $q$-tree as a spanning subgraph, then the chromatic polynomial $P(G,\lambda)$ of $G$ has no non-integer zeros in the interval $(0,q)$. Sokal conjectured that for any graph $G$ and any real $\lambda>\Delta(G)$, $P(G,\lambda)>0$. Our second result confirms that it is true if $\Delta(G)\ge \lfloor n/3\rfloor -1$, where $n$ is the order of $G$.