  This work is devoted to P\'olya-Young urns, a class of periodic P\'olya urns
of importance in the analysis of Young tableaux. We provide several extension
of the previous results of Banderier, Marchal and Wallner [Ann. Prob. (2020)]
on P\'olya-Young urns and also generalize the previously studied model. We
determine the limit law of the generalized model, involving the the local time
of noise-reinforced Bessel processes. We also uncover a martingale structure,
which leads directly to almost-sure convergence of the random variable of
interest. This allows us to add second order asymptotics by providing a central
limit theorem for the martingale tail sum, as well as a law of the iterated
logarithm. We also turn to random vectors and obtain the limit law of
P\'olya-Young urns with multiple colors. Additionally, we introduce several
growth processes and combinatorial objects, which are closely related to urn
models. We define increasing trees with periodic immigration and we related the
dynamics of the P\'olya-Young urns to label-based parameters in such tree
families. Furthermore, we discuss a generalization of Stirling permutations and
obtain a bijection to increasing trees with periodic immigration. Finally, we
introduce a chinese restaurant process with competition and relate it to
increasing trees, as well as P\'olya-Young urns.
