Growth of Cayley and diluted Cayley trees with two kinds of entities

A kinetic growth model derived from the magnetic Eden model is introduced in order to simulate the growth of hierarchical structures, such as Cayley trees. We only consider the case where two kinds of entities are competing with each other can be further subjected to an external field. The very relevant case in which both kinds of entities have different coordination numbers is introduced here for the first time, and is called the diluted Cayley tree. Physical and geometrical properties of the finite and infinite trees are exactly found and simulated. Finite-size effects are emphasized and illustrated on the global or local magnetization and on the chemical activity. Asymptotic limits are given in each case. The generated patterns can be related to a correlated percolation problem briefly discussed in the appendix.