\( \newcommand{\depth}{\textnormal{depth}} \newcommand{\height}{\textnormal{height}} \newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} \newcommand{\ceil}[1]{\left\lceil #1 \right\rceil} \DeclareMathOperator*{\E}{\mathbb{E}} \newcommand{\pfrac}[2]{\left(\frac{#1}{#2}\right)} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \)