  By introducing the notions of living and dead nodes a new model of random
tree evolution with continuous time parameter has been constructed. It is
assumed that two random variables, the lifetime and the offspring number of
living nodes control the evolution process. It has been shown that the
generating function of the probability to find the evolving tree in a given
state satisfies a non-linear integral equation. Analyzing the time dependence
of the average number of living nodes three different types of evolution
(subcritical, critical and supercritical) can be observed. It has been proved
that in the case of subcritical evolution there is a well-defined time point at
which the variance of the number of living nodes has a maximum. The joint
distribution function of the numbers of living and dead nodes has been derived,
and the time dependence of the correlation between these node numbers has been
calculated. It is found that in the critical evolution with increasing time the
correlation converges to a fixed number which is independent of the
distributions of the lifetime and the offspring number of living nodes.
