In this talk, Wwe will present a relation between weighted Catalan numbers and the phase transitions of a competitive growth process on a d-ary tree. The competitive growth processes presenting this relation are generalizations of chase-escape: chase-escape with death and distance-dependent chase-escape, which mimic the dynamics of a predator chasing prey, and the latter have random lifespans.

This talk is based on joint works with Erin Beckman, Keisha Cook, Nicole Eikmeier, and Matthew Junge; and with Matthew Junge, Naina Ray and Nidhi Ray.