  A particular continuous-time multitype branching process is considered, it is
the continuous-time embedding of a discrete-time process which is very popular
in theoretical computer science: the m-ary search tree (m is an integer). There
is a well-known phase transition: when m \leq 26, the asymptotic behavior of
the process is Gaussian, but for m \geq 27 it is no more Gaussian and a limit W
of a complex-valued martingale arises. Thanks to the branching property it
appears as a solution of a smoothing equation of the type Z =
e^{-{\lambda}T}(Z(1) + ... + Z(m)), where {\lambda} \in C, the Z(k) are
independent copies of Z and T is a R_+-valued random variable, independent of
the Z(k). This distributional equation is extensively studied by various
approaches. The existence and unicity of solution of the equation are proved by
contraction methods. The fact that the distribution of W is absolutely
continuous and that its support is the whole complex plane is shown via Fourier
analysis. Finally, the existence of exponential moments of W is obtained by
considering W as the limit of a complex Mandelbrot cascade.
