We compute, via motivic wall-crossing, the generating function of virtual motives of the Quot scheme of points on ${\mathbb{A}}^3$ , generalising to higher rank a result of Behrend–Bryan–Szendrői. We show that this motivic partition function converges to a Gaussian distribution, extending a result of Morrison.

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A fringe subtree of a rooted tree is a subtree that consists of a node and all its descendants. In this paper, we are particularly interested in the number of fringe subtrees that occur repeatedly in a random rooted tree. Specifically, we show that the average number of fringe subtrees that occur at least r times is of asymptotic order n/(log n) 3/2 for every r ≥ 2 (with small periodic fluctuations in the main term) if a tree is taken uniformly at random from a simply generated family. Moreover, we also prove a strong concentration result for a related parameter: the size of the smallest tree that does not occur as a fringe subtree is with high probability equal to one of at most two different values. The main proof ingredients are singularity analysis, bootstrapping and the first and second moment methods.

A tree functional is called additive if it satisfies a recursion of the form F (T ) = k j=1 F (Bj) + f (T ), where B1, . . . , B k are the branches of the tree T and f (T ) is a toll function. We prove a general central limit theorem for additive functionals of d-ary increasing trees under suitable assumptions on the toll function. The same method also applies to generalised plane-oriented increasing trees (GPORTs). One of our main applications is a log-normal law that we prove for the size of the automorphism group of d-ary increasing trees, but many other examples (old and new) are covered as well.

An additive functional of a rooted tree is a functional that can be calculated recursively as the sum of the values of the functional over the branches, plus a certain toll function. Janson recently proved a central limit theorem for additive functionals of conditioned Galton-Watson trees under the assumption that the toll function is local, i.e. only depends on a fixed neighbourhood of the root. We extend his result to functionals that are "almost local" in a certain sense, thus covering a wider range of functionals. The notion of almost local functional intuitively means that the toll function can be approximated well by considering only a neighbourhood of the root. Our main result is illustrated by several explicit examples including natural graph theoretic parameters such as the number of independent sets, the number of matchings, and the number of dominating sets. We also cover a functional stemming from a tree reduction process that was studied