  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.
