Stephan Georg Wagner

Uppsala University | UU · Department of Mathematics

The protection number of a vertex $v$ in a tree is the length of the shortest path from $v$ to any leaf contained in the maximal subtree where $v$ is the root. In this paper, we determine the distribution of the maximum protection number of a vertex in simply generated trees, thereby refining a recent result of Devroye, Goh, and Zhao. Two different...

For a complex number $$\alpha $$ α , we consider the sum of the $$\alpha $$ α th powers of subtree sizes in Galton–Watson trees conditioned to be of size n . Limiting distributions of this functional $$X_n(\alpha )$$ X n ( α ) have been determined for $${\text {Re}}\alpha \ne 0$$ Re α ≠ 0 , revealing a transition between a complex normal limiting d...

For a $k$-tree $T$, we prove that the maximum local mean order is attained in a $k$-clique of degree $1$ and that it is not more than twice the global mean order. We also bound the global mean order if $T$ has no $k$-cliques of degree $2$ and prove that for large order, the $k$-star attains the minimum global mean order. These results solve the rem...

A subdiagonal composition of a positive integer is a composition with the property that the ith part is less than or equal to i, and analogously a superdiagonal composition is a composition with the property that the ith part is greater than or equal to i. The generating functions for subdiagonal and superdiagonal compositions as well as for compos...

Let n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document} be an integer. We prove the convexity of the so-called MacMahon q-Catalan polynomials Cn(q...

We consider the process of uncovering the vertices of a random labeled tree according to their labels. First, a labeled tree with n vertices is generated uniformly at random. Thereafter, the vertices are uncovered one by one, in order of their labels. With each new vertex, all edges to previously uncovered vertices are uncovered as well. In this wa...

We study the size of the automorphism group of two different types of random trees: Galton–Watson trees and rooted Pólya trees. In both cases, we prove that it asymptotically follows a log-normal distribution and provide asymptotic formulas for the mean and variance of the logarithm of the size of the automorphism group. While the proof for Galton–...

It is known that random directed graphs D(n,p)$$ D\left(n,p\right) $$ undergo a phase transition around the point p=1/n$$ p=1/n $$. Earlier, Łuczak and Seierstad have established that as n→∞$$ n\to \infty $$ when p=(1+μn−1/3)/n$$ p=\left(1+\mu {n}^{-1/3}\right)/n $$, the asymptotic probability that the strongly connected components of a random dire...

The protection number of a vertex $v$ in a tree is the length of the shortest path from $v$ to any leaf contained in the maximal subtree where $v$ is the root. In this paper, we determine the distribution of the maximum protection number of a vertex in simply generated trees, thereby refining a recent result of Devroye, Goh and Zhao. Two different...

A k - plane tree is a plane tree whose vertices are assigned labels between 1 and k in such a way that the sum of the labels along any edge is no greater than $$k+1$$ k + 1 . These trees are known to be related to $$(k+1)$$ ( k + 1 ) -ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple ref...

We consider random two‐colorings of random linear preferential attachment trees, which includes recursive trees, plane‐oriented recursive trees, binary search trees, and a class of d‐ary trees. The random coloring is defined by assigning the root the color red or blue with equal probability, and all other vertices are assigned the color of their pa...

For fixed non-negative integers $k$, $t$, and $n$, with $t < k$, a $k_t$-Dyck path of length $(k+1)n$ is a lattice path that starts at $(0, 0)$, ends at $((k+1)n, 0)$, stays weakly above the line $y = -t$, and consists of steps from the step-set $\{(1, 1), (1, -k)\}$. We enumerate the family of $k_t$-Dyck paths by considering the number of down-ste...

The Gini index of a set partition π of size n is defined as 1 − δ(π)/ n^2 , where δ(π) is the sum of the squares of the block cardinalities of π. In this paper, we study the distribution of the δ statistic on various kinds of set partitions in which the first r elements are required to lie in distinct blocks. In particular, we derive the generating...

We consider the process of uncovering the vertices of a random labeled tree according to their labels. First, a labeled tree with $n$ vertices is generated uniformly at random. Thereafter, the vertices are uncovered one by one, in order of their labels. With each new vertex, all edges to previously uncovered vertices are uncovered as well. In this...

The class of linear preferential attachment trees includes recursive trees, plane-oriented recursive trees, binary search trees, and increasing $d$-ary trees. Bond percolation with parameter $p$ is performed by considering every edge in a graph independently, and either keeping the edge with probability $p$ or removing it otherwise. The resulting c...

For a complex number $\alpha$, we consider the sum of the $\alpha$th powers of subtree sizes in Galton--Watson trees conditioned to be of size $n$. Limiting distributions of this functional $X_n(\alpha)$ have been determined for $\Re\alpha \neq 0$, revealing a transition between a complex normal limiting distribution for $\Re\alpha < 0$ and a non-n...

We study the size of the automorphism group of two different types of random trees: Galton--Watson trees and rooted P\'olya trees. In both cases, we prove that it asymptotically follows a log-normal distribution and provide asymptotic formulas for mean and variance of the logarithm of the size of the automorphism group. While the proof for Galton--...

The problem of determining the maximum number of maximal independent sets in certain graph classes dates back to a paper of Miller and Muller and a question of Erd\H{o}s and Moser from the 1960s. The minimum was always considered to be less interesting due to simple examples such as stars. In this paper we show that the problem becomes interesting...

The Gini index of a set partition π of size n is defined as 1 − δ ( π ) n 2 , $1-\frac{\delta (\pi )}{{{n}^{2}}},$ where δ ( π ) is the sum of the squares of the block cardinalities of π . In this paper, we study the distribution of the δ statistic on various kinds of set partitions in which the first r elements are required to lie in distinct bloc...

We characterize the extremal trees that maximize the number of almost-perfect matchings, which are matchings covering all but one or two vertices, and those that maximize the number of strong almost-perfect matchings, which are matchings missing only one or two leaves. We also determine the trees that minimize the number of maximal matchings. We ap...

For a tree T $T$, the mean subtree order of T $T$ is the average order of a subtree of T $T$. In 1984, Jamison conjectured that the mean subtree order of T $T$ decreases by at least 1/3 after contracting an edge in T $T$. In this article we prove this conjecture in the special case that the contracted edge is a pendant edge. From this result, we ha...

For a tree T , the mean subtree order of T is the average order of a subtree of T. In 1984, Jamison conjectured that the mean subtree order of T decreases by at least 1/3 after contracting an edge in T. In this paper we prove this conjecture in the special case that the contracted edge is a pendant edge. From this result, we have a new proof of the...

We prove that the multiplicity of a fixed eigenvalue $\alpha$ in a random recursive tree on $n$ vertices satisfies a central limit theorem with mean and variance asymptotically equal to $\mu_{\alpha} n$ and $\sigma^2_{\alpha} n$ respectively. It is also shown that $\mu_{\alpha}$ and $\sigma^2_{\alpha}$ are positive for every totally real algebraic...

A fringe subtree of a rooted tree is a subtree induced by one of the vertices and all its descendants. We consider the problem of estimating the number of distinct fringe subtrees in random trees under a generalized notion of distinctness, which allows for many different interpretations of what “distinct” trees are. The random tree models considere...

A $k$-plane tree is a plane tree whose vertices are assigned labels between $1$ and $k$ in such a way that the sum of the labels along any edge is no greater than $k+1$. These trees are known to be related to $(k+1)$-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting form...

Let T be a rooted tree, and V(T) its set of vertices. A subset X of V(T) is called an infima closed set of T if for any two vertices u,v∈X, the first common ancestor of u and v is also in X. This paper determines the trees with minimum number of infima closed sets among all rooted trees of given order, thereby answering a question of Klazar. It is...

For fixed non-negative integers $k$, $t$, and $n$, with $t < k$, a $k_t$-Dyck path of length $(k+1)n$ is a lattice path that starts at $(0, 0)$, ends at $((k+1)n, 0)$, stays weakly above the line $y = -t$, and consists of steps from the step-set $\{(1, 1), (1, -k)\}$. We enumerate the family of $k_t$-Dyck paths by considering the number of down-ste...

For a graph G, we denote by N(G) the number of non-empty subtrees of G. As a topological index based on counting, N(G) has some correlations to other well studied topological indices, including the Wiener index W(G). In this paper we characterize the extremal graphs with the maximum number of subtrees among all cacti of order n with k cycles. Simil...

For a graph G, we denote by N (G) the number of non-empty subtrees of G. As a topological index based on counting, N (G) has some correlations to other well studied topological indices, including the Wiener index W (G). In this paper we characterize the extremal graphs with the maximum number of subtrees among all cacti of order n with k cycles. Si...

Phylogenetic trees are used to model evolution: leaves are labelled to represent contemporary species ("taxa") and interior vertices represent extinct ancestors. Informally, convex characters are measurements on the contemporary species in which the subset of species (both contemporary and extinct) that share a given state, form a connected subtree...

In this work we consider random two-colourings of random linear preferential attachment trees, which includes random recursive trees, random plane-oriented recursive trees, random binary search trees, and a class of random $d$-ary trees. The random colouring is defined by assigning the root of the tree the colour red or blue with equal probability,...

A subtree of a tree is any induced subgraph that is again a tree (i.e., connected). The mean subtree order of a tree is the average number of vertices of its subtrees. This invariant was first analyzed in the 1980s by Jamison. An intriguing open question raised by Jamison asks whether the maximum of the mean subtree order, given the order of the tr...

A fringe subtree of a rooted tree is a subtree induced by one of the vertices and all its descendants. We consider the problem of estimating the number of distinct fringe subtrees in two types of random trees: simply generated trees and families of increasing trees (recursive trees, $d$-ary increasing trees and generalized plane-oriented recursive...

We consider the quantity P ( G ) associated with a graph G that is defined as the probability that a randomly chosen subtree of G is spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of this graph invariant depending on the edge density, we establish first that P ( G ) is bounded below by a positive consta...

The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for exampl...

A fringe subtree of a rooted tree is a subtree consisting of one of the nodes and all its descendants. In this paper, we are specifically interested in the number of non-isomorphic trees that appear in the collection of all fringe subtrees of a binary tree. This number is analysed under two different random models: uniformly random binary trees and...

A subtree of a tree is any induced subgraph that is again a tree (i.e., connected). The mean subtree order of a tree is the average number of vertices of its subtrees. This invariant was first analyzed in the 1980s by Jamison. An intriguing open question raised by Jamison asks whether the maximum of the mean subtree order, given the order of the tr...

Random directed graphs $D(n,p)$ undergo a phase transition around the point $p = 1/n$, and the width of the transition window has been known since the works of Luczak and Seierstad. They have established that as $n \to \infty$ when $p = (1 + \mu n^{-1/3})/n$, the asymptotic probability that the strongly connected components of a random directed gra...

Two subclasses of Motzkin paths, S-Motzkin and T-Motzkin paths, are introduced. We provide bijections between S-Motzkin paths and ternary trees, S-Motzkin paths and non-crossing trees, and T-Motzkin paths and ordered pairs of ternary trees. Symbolic equations for both paths, and thus generating functions for the paths, are provided. Using these, va...

Let $T$ be a rooted tree, and $V(T)$ its set of vertices. A subset $X$ of $V(T)$ is called an infima closed set of $T$ if for any two vertices $u,v\in X$, the first common ancestor of $u$ and $v$ is also in $X$. This paper determines the trees with minimum number of infima closed sets among all rooted trees of given order, thereby answering a quest...

The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for exampl...

We study the average size of independent (vertex) sets of a graph. This invariant can be regarded as the logarithmic derivative of the independence polynomial evaluated at 1. We are specifically concerned with extremal questions. The maximum and minimum for general graphs are attained by the empty and complete graph respectively, while for trees we...

In this paper, we consider the average size of independent edge sets, also called matchings, in a graph. We characterize the extremal graphs for the average size of matchings in general graphs and trees. In addition, we obtain inequalities between the average size of matchings and the number of matchings as well as the matching energy, which is def...

We consider integer sequences that satisfy a recursion of the form $x_{n+1} = P(x_n)$ for some polynomial $P$ of degree $d > 1$. If such a sequence tends to infinity, then it satisfies an asymptotic formula of the form $x_n \sim A \alpha^{d^n}$, but little can be said about the constant $\alpha$. In this paper, we show that $\alpha$ is always irrat...

A fringe subtree of a rooted tree is a subtree consisting of one of the nodes and all its descendants. In this paper, we are specifically interested in the number of non-isomorphic trees that appear in the collection of all fringe subtrees of a binary tree. This number is analysed under two different random models: uniformly random binary trees and...

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. Svante Janson recently proved a central limit theorem for additive functionals of conditioned Galton–Watson trees under the assumption that the toll function is local,...

Let m(G,k) denote the number of matchings of cardinality k in a graph G. A quasi-order ⪯ is defined by writing G⪯H whenever m(G,k)≤m(H,k) holds for all k. We consider the set G1(n,γ) of connected graphs with n vertices and γ cut vertices as well as the set G2(n,γ) of connected graphs with n vertices and γ cut edges. We determine the greatest and le...

We study the asymptotic number of certain monotonically labeled increasing trees arising from a generalized evolution process. The main difference between the presented model and the classical model of binary increasing trees is that the same label can appear in distinct branches of the tree. In the course of the analysis we develop a method to ext...

We consider the quantity $P(G)$ associated with a graph $G$ that is defined as the probability that a randomly chosen subtree of $G$ is spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of this graph invariant depending on the edge density, we establish first that $P(G)$ is bounded below by a positive cons...

Given a rooted tree T with leaves v 1 ,v 2 ,…,v n , we define the ancestral matrix C(T) of T to be the n×n matrix for which the entry in the i-th row, j-th column is the level (distance from the root) of the first common ancestor of v i and v j . We study properties of this matrix, in particular regarding its spectrum: we obtain several upper and l...

Given a finite set of bases b1, b2, …, br (integers greater than 1), a multi-base representation of an integer n is a sum with summands db1α1b2α2⋯brαr, where the αj are nonnegative integers and the digits d are taken from a fixed finite set. We consider multi-base representations with at least two bases that are multiplicatively independent. Our ma...

A composition of the positive integer n is a representation of n as an ordered sum of positive integers (Formula presented.) There are (Formula presented.) unrestricted compositions of n, which can be sorted according to the number of inversions they contain. (An inversion in a composition is a pair of summands (Formula presented.) for which (Formu...

In this paper, we consider the average size of independent edge sets, also called matchings, in a graph. We characterize the extremal graphs for the average size of matchings in general graphs and trees. In addition, we obtain inequalities between the average size of matchings and the number of matchings as well as the matching energy, which is def...

A tree functional is called additive if it satisfies a recursion of the form $F(T)=\sum_{j=1}^k F(B_j)+f(T)$ , where B1 ,…, Bk 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 al...

Two subclasses of Motzkin paths, S-Motzkin and T-Motzkin paths, are introduced. We provide bijections between S-Motzkin paths and ternary trees, S-Motzkin paths and non-crossing trees, and T-Motzkin paths and ordered pairs of ternary trees. Symbolic equations for both paths, and thus generating functions for the paths, are provided. Using these, va...

We consider the problem of maximizing the distance spectral radius and a slight generalization thereof among all trees with some prescribed degree sequence. We prove in particular that the maximum of the distance spectral radius has to be attained by a caterpillar for any given degree sequence. The same holds true for the terminal distance matrix....

The quantity that captures the asymptotic value of the maximum number of appearances of a given topological tree (a rooted tree with no vertices of outdegree $1$) $S$ with $k$ leaves in an arbitrary tree with sufficiently large number of leaves is called the inducibility of $S$. Its precise value is known only for some specific families of trees, m...

For a $d$-ary tree (every vertex has outdegree between $2$ and $d$) $D$ with $|D|=k$ leaves, let $\gamma(D,T)$ be the density of all subsets of $k$ leaves of the $d$-ary tree $T$ that induce a copy of $D$. The inducibility of $D$ is $\limsup_{|T|\to \infty}\gamma(D,T)$. We give a general upper bound on the inducibility of $D$ as a function of the i...

For a $d$-ary tree (every vertex has outdegree between $2$ and $d$) $D$ with $|D|=k$ leaves, let $\gamma(D,T)$ be the density of all subsets of $k$ leaves of the $d$-ary tree $T$ that induce a copy of $D$. The inducibility of $D$ is $\limsup_{|T|\to \infty}\gamma(D,T)$. We give a general upper bound on the inducibility of $D$ as a function of the i...

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. o...

Given a rooted tree $T$ with leaves $v_1,v_2,\ldots,v_n$, we define the ancestral matrix $C(T)$ of $T$ to be the $n \times n$ matrix for which the entry in the $i$-th row, $j$-th column is the level (distance from the root) of the first common ancestor of $v_i$ and $v_j$. We study properties of this matrix, in particular regarding its spectrum: we...

Given a finite set of bases $b_1$, $b_2$, \dots, $b_r$ (integers greater than $1$), a multi-base representation of an integer~$n$ is a sum with summands $db_1^{\alpha_1}b_2^{\alpha_2} \cdots b_r^{\alpha_r}$, where the $\alpha_j$ are nonnegative integers and the digits $d$ are taken from a fixed finite set. We consider multi-base representations wit...

We consider a procedure to reduce simply generated trees by iteratively removing all leaves. In the context of this reduction, we study the number of vertices that are deleted after applying this procedure a fixed number of times by using an additive tree parameter model combined with a recursive characterization. Our results include asymptotic for...

In this paper, we study the average size of independent (vertex) sets of a graph. This invariant can be regarded as the logarithmic derivative of the independence polynomial evaluated at $1$. We are specifically concerned with extremal questions. The maximum and minimum for general graphs are attained by the empty and complete graph respectively, w...

Belief propagation (BP) has been applied in a variety of inference problems as an approximation tool. BP does not necessarily converge in loopy graphs, and even if it does, is not guaranteed to provide exact inference. Even so, BP is useful in many applications due to its computational tractability. In this article, we investigate a regularized BP...

We give a lower bound for the number of total dominating sets of a graph together with a characterization of the extremal graphs, for trees as well as arbitrary connected graphs of given order. Moreover, we obtain a sharp lower bound involving both the order and the total domination number, and characterize the extremal graphs as well.

We investigate a Tutte-like polynomial for rooted trees and posets called $\mathcal{V}$-posets. These posets are obtained recursively by either disjoint unions or adding a greatest/least element to existing $\mathcal{V}$-posets, and they can also be characterised as those posets that do not contain an $N$-poset or a bowtie as induced subposets. We...

A centroid node in a tree is a node for which the sum of the distances to all other nodes attains its minimum, or equivalently a node with the property that none of its branches contains more than half of the other nodes. We generalise some known results regarding the behaviour of centroid nodes in random recursive trees (due to Moon) to the class...

Trees without vertices of degree $2$ are sometimes named topological trees. In this work, we bring forward the study of the inducibility of (rooted) topological trees with a given number of leaves. The inducibility of a topological tree $S$ is the limit superior of the proportion of all subsets of leaves of $T$ that induce a copy of $S$ as the size...

Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of $d$-ary trees (rooted trees whose vertex outdegrees are bounded from above by $d\geq 2$) with a given number of leaves. We determine the exact inducibility for stars and binary caterpillars. For $T$ in the family of strictly $d$-ary trees (every vertex...

A nonempty word w of finite length over the alphabet of positive integers is a Stirling word if for each letter i in w all entries between two consecutive occurrences of i (if these exist) are larger or equal to i. We derive an exact and also an asymptotic formula for the probability that a random geometrically distributed word of length n is a Sti...

We investigate the distribution of the number of vertices of a randomly chosen subtree of a tree. Specifically, it is proven that this distribution is close to a Gaussian distribution in an explicitly quantifiable way if the tree has sufficiently many leaves and no long branchless paths. We also show that the conditions are satisfied asymptotically...

We consider point spectrum traces in the Hofstadter model. We show how to recover the full quantum Hofstadter trace by integrating these point spectrum traces with the appropriate free density of states on the lattice. This construction is then generalized to the almost Mathieu operator and its n-th moments which can be expressed in terms of genera...

The Turán connected graph \(\mathrm {TC}_{n,\alpha }\) is obtained from \(\alpha \) cliques of size \(\lfloor \frac{n}{\alpha } \rfloor \) or \(\lceil \frac{n}{\alpha } \rceil \) by joining all cliques by an edge to one central vertex in one of the larger cliques. The graph \(\mathrm {TC}_{n,\alpha }\) was shown recently by Bruyère and Mélot to max...

Given an integer $k \geq 2$ and a real number $\gamma\in [0, 1]$, which graphs of edge density $\gamma$ contain the largest number of $k$-edge stars? For $k=2$ Ahlswede and Katona proved that asymptotically there cannot be more such stars than in a clique or in the complement of a clique (depending on the value of $\gamma$). Here we extend their re...

A tanglegram consists of two rooted binary plane trees with the same number of leaves and a perfect matching between the two leaf sets. Tanglegrams are drawn with the leaves on two parallel lines, the trees on either side of the strip created by these lines, and the perfect matching inside the strip. If this can be done without any edges crossing,...

A Lucas sequence is a sequence of the general form \(v_{n} = (\phi ^{n} -\overline{\phi }^{n})/(\phi -\overline{\phi })\), where ϕ and \(\overline{\phi }\) are real algebraic integers such that \(\phi +\overline{\phi }\) and \(\phi \overline{\phi }\) are both rational. Famous examples include the Fibonacci numbers, the Pell numbers, and the Mersenn...

A segment of a tree is a path whose end vertices have degree 1 or at least 3, while all internal vertices have degree 2. The lengths of all the segments of form its segment sequence, in analogy to the degree sequence. We address a number of extremal problems for the class of all trees with a given segment sequence. In particular, we determine the e...

We construct minor-closed addable families of graphs that are subcritical and contain all planar graphs. This contradicts (one direction of) a well-known conjecture of Noy.

Bootstrap percolation is a growth model inspired by cellular automata. At the initial time \(t=0\), the bootstrap percolation process starts from an initial random configuration of active vertices on a given graph, and proceeds deterministically so that a node becomes active at time \(t=1,2,\dots \) if sufficiently many of its neighbors are already...

Consider non-negative lattice paths ending at their maximum height, which will be called admissible paths. We show that the probability for a lattice path to be admissible is related to the Chebyshev polynomials of the first or second kind, depending on whether the lattice path is defined with a reflective barrier or not. Parameters like the number...

Betweenness centrality is a quantity that is frequently used to measure how ‘central’ a vertex v is. It is defined as the sum, over pairs of vertices other than v, of the proportions of shortest paths that pass through v. In this paper, we study the distribution of the betweenness centrality in random trees and related, subcritical graph families....

We consider the generating function of the algebraic area of lattice walks, evaluated at a root of unity, and its relation to the Hofstadter model. In particular, we obtain an expression for the generating function of the n-th moments of the Hofstadter Hamiltonian in terms of a complete elliptic integral, evaluated at a rational function. This in t...

In this paper, we introduce the notion of $q$-quasiadditivity of arithmetic functions, as well as the related concept of $q$-quasimultiplicativity, which generalise strong $q$-additivity and -multiplicativity, respectively. We show that there are many natural examples for these concepts, which are characterised by functional equations of the form $...

We use analytic methods to study the probability of a family of motifs not occurring on the fringe of a random recursive tree. We obtain an asymptotic formula for this probability by means of singularity analysis. Two regimes are treated in particular: the case that a fixed proportion of motifs of size γ is forbidden, and the case that a fixed numb...

A Lucas sequence is a sequence of the general form $v_n = (\phi^n - \bar{\phi}^n)/(\phi-\bar{\phi})$, where $\phi$ and $\bar{\phi}$ are real algebraic integers such that $\phi+\bar{\phi}$ and $\phi\bar{\phi}$ are both rational. Famous examples include the Fibonacci numbers, the Pell numbers, and the Mersenne numbers. We study the monoid that is gen...

Multi-edge trees as introduced in a recent paper of Dziemia\'nczuk are plane trees where multiple edges are allowed. We first show that $d$-ary multi-edge trees where the out-degrees are bounded by $d$ are in bijection with classical $d$-ary trees. This allows us to analyse parameters such as the height. The main part of this paper is concerned wit...

A tree functional is called additive if it satisfies a recursion of the form $F(T) = \sum_{j=1}^k F(B_j) + f(T)$, where $B_1,\ldots,B_k$ 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 sa...

In this paper, we introduce the notion of $q$-quasiadditivity of arithmetic functions, as well as the related concept of $q$-quasimultiplicativity, which generalises strong $q$-additivity and -multiplicativity, respectively. We show that there are many natural examples for these concepts, which are characterised by functional equations of the form...

In this chapter we explore recent development on various problems related to graph indices in trees. We focus on indices based on distances between vertices, vertex degrees, or on counting vertex or edge subsets of different kinds. Some of the indices arise naturally in applications, e.g., in chemistry, statistical physics, bioinformatics, and othe...