We examine the question of solving the extinction

probability of a particular class of continuous-time multi-type

branching processes, named Markovian binary trees (MBT). The

extinction probability is the minimal nonnegative solution of a

fixed point equation that turns out to be quadratic, which makes its

resolution particularly clear.

We analyze first two linear algorithms to compute the extinction

probability of an MBT, of which one is new, and, we propose a

quadratic algorithm arising from Newton's iteration method for

fixed-point equations.

Finally, we add a catastrophe process to the

initial MBT, and we analyze the resulting system. The extinction

probability turns out to be much more difficult to compute; we use a

$G/M/1$-type Markovian process approach to approximate this

probability.