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The space requirements of an $m$-ary search tree satisfies a well-known phasetransition: 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-valuedmartingale arises. We show that the distribution of $W$ has a square integrabledensity on the complex plane, that its support is the whole complex plane, andthat it has finite exponential moments. The proofs are based on the study ofthe 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 variablesuniformly distributed on $[0,1]$, $W_1, ..., W_m$ are independent copies of Wwhich are also independent of $(V_1, ..., V_m)$ and $\lambda$ is a complexnumber.

Support and density of the limit $m$-ary search trees distribution