Polynomials with many rational preperiodic points

In this paper we study two questions related to exceptional behavior of preperiodic points of polynomials in $\mathbb{Q}[x]$. We show that for all $d\ge 2$, there exists a polynomial $f_d(x)\in \mathbb{Q}[x]$ with $2\leq $\mathrm{d}\mathrm{e}\mathrm{g}$(f_d) \leq d$suchthat$f_d(x)$hasatleast$d + \lfloor \log_2(d)\rfloor$ rational preperiodic points. Furthermore, we show that for infinitely many integers $d$, the polynomials $f_d(x)$ and $f_d(x)+1$ have at least $d^2+d\lfloor {\log }_2(d)\rfloor -2d+1$ common complex preperiodic points.