This paper extends the study of fringe trees in random plane trees with a given degree statistic. While previous work established the asymptotic normality of the count of fringe trees isomorphic to a fixed tree, we investigate the case where the target tree grows with the size of the random tree. We consider three primary subtree counts: the number of fringe trees isomorphic to a specific growing tree, the number of fringe trees sharing a given growing degree statistic, and the number of fringe trees of a specific growing size. To establish our results, we employ and compare four distinct probabilistic frameworks: the method of moments with the Gao-Wormald theorem, Stein's method with coupling (to provide explicit error bounds in total variation distance), the Cai-Devroye method, and Stein's method with exchangeable pairs. Our findings provide conditions for Poisson and normal convergence for these subtree counts. Additionally, we provide a local limit theorem for sums of values obtained via sampling without replacement that may be of independent interest. Finally, our results and methods are also applied to conditioned critical Galton-Watson trees.