  For simplest colored branching processes we prove an analog to the McMillan
theorem and calculate Hausdorff dimensions of random fractals defined in terms
of the limit behavior of empirical measures generated by finite genetic lines.
In this setting the role of Shannon's entropy is played by the
Kullback--Leibler divergence and the Hausdorff dimensions are computed by means
of the so-called Billingsley--Kullback entropy, defined in the paper.
