  The space requirements of an $m$-ary search tree satisfies a well-known phase
transition: when $m\leq 26$, the second order asymptotics is Gaussian. When
$m\geq 27$, it is not Gaussian any longer and a limit $W$ of a complex-valued
martingale arises. We show that the distribution of $W$ has a square integrable
density on the complex plane, that its support is the whole complex plane, and
that it has finite exponential moments. The proofs are based on the study of
the distributional equation $ W\egalLoi\sum_{k=1}^mV_k^{\lambda}W_k$, where
$V_1, ..., V_m$ are the spacings of $(m-1)$ independent random variables
uniformly distributed on $[0,1]$, $W_1, ..., W_m$ are independent copies of W
which are also independent of $(V_1, ..., V_m)$ and $\lambda$ is a complex
number.
