On properties of branching processes with an arbitrary set of types of particles

A method of analysis of branching processes using the space of tree sections (the tree is the set of offsprings with their genealogies) is presented. Some formulas and theorems stated previously for particular sets are generalized to the case of an arbitrary set of types of particles. Some results concerning the maximal distance of relationship are given. The existence of a non-degenerate limit distribution for the maximal distance of relationship in a group with a fixed number of offsprings is proved. This distribution is estimated for some particular cases.