On some properties of Perron numbers

Let $\theta$ be a real number, $n\in \mathbb{N}$, and $D_n(\theta )=\{{\sum }_{k=1}^n{a}_k{\theta }^k\mid a_k\in \{0,\dots ,\lfloor \theta \rfloor \}\}\mathrm{.}$ Let $\theta$ be a Perron number, that is, an algebraic integer $>1$ whose other Galois conjugates are less than $\theta$ in absolute value. I shall prove two results: (1) ${\theta }^n\ll \mathrm{\#}{D}_n(\theta )\ll \sqrt{n}{\theta }^n\mathrm{.}$ (2) $\theta$ is of height $\le \lfloor \theta \rfloor$.