Oscillations in the height of the Yule tree and application to the binary search tree

For a particular case of a branching random walk with lattice support, namely the Yule branching random walk, we prove that the distribution of the centred maximum oscillates around a distribution corresponding to a critical travelling wave in the following sense: there exist continuous functions $t \mapsto a_t$ and $x \mapsto \overline{\phi}(x)$ such that: $$\lim_{t \rightarrow +\infty} \sup_{x \in \mathbb{R}} \vert \mathbb{P}(\overline{X}(t) \leq a_t +x )-\overline{\phi}(x- \{ a_t +x\})\vert=0,$$ where $\{x\}=x-\lfloor x \rfloor$ and $\overline{X}(t)$ is the height of the Yule tree. We also shows that similar oscillations occur for $\mathbb{E}\left(f(\overline{X}(t)-a_t)\right)$, when $f$ is in a large class of functions. This process is classically related to the binary search tree, thus yielding analogous results for the height and for the saturation level of the binary search tree.