Rare subtree patterns in size-conditioned Bienaym\'e trees: Poisson approximation and declumping

Abstract

We establish a general Poisson approximation for rare local patterns in critical Bienaym\'e--Galton--Watson trees with offspring distribution $\mu$ in the domain of attraction of a stable law, conditioned to have a large number of vertices. A pattern is specified by a sequence-dependent mark on fringe subtrees. If marked fringe subtrees remain microscopic and nearby marked occurrences have negligible clustering, then their count is asymptotically Poisson in total variation whenever its mean remains bounded; when the mean diverges, the count satisfies a law of large numbers. The main difficulty is the global dependence created by size conditioning. We overcome it by combining the cyclic-shift representation with a refined form of the Chen--Stein bound and a bridge-removal estimate controlling the interaction between a local mark and the remainder of the conditioned random walk. For non-fringe patterns, overlapping occurrences may form clusters and the raw count need not be asymptotically Poisson. We introduce declumped indicators which select boundary witnesses of these clusters and prove a general Poisson approximation for their count. As applications, we obtain sharp asymptotics for the maximum leaf-height, equivalently the maximum protection number, and for the height of the largest complete $r$-ary tree appearing as a non-fringe subtree. Unary-chain maxima, and the maximum leaf-height when $\mu_1>0$, exhibit lattice-modulated Gumbel behavior. Complete $r$-ary patterns for $r\ge2$, and the maximum leaf-height when $\mu_1=0$, are localized on one or two consecutive integers. The results require no exponential moment and include offspring distributions with infinite variance.