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It follows from a classical result of Jordan that every tree with maximum degree at most $r$ containing a vertex set labeled by $[n]$, has a single-edge cut which separates two subsets $A,B \subset [n]$ for which $\min\{|A|,|B|\} \ge…
We study the effectiveness of subagging, or subsample aggregating, on regression trees, a popular non-parametric method in machine learning. First, we give sufficient conditions for pointwise consistency of trees. We formalize that (i) the…
We study the additive functional $X_n(\alpha)$ on conditioned Galton-Watson trees given, for arbitrary complex $\alpha$, by summing the $\alpha$th power of all subtree sizes. Allowing complex $\alpha$ is advantageous, even for the study of…
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…
The basic object we consider is a certain model of continuum random tree, called the stable tree. We construct a fragmentation process $(F^-(t), t>=0)$ out of this tree by removing the vertices located under height $t$. Thanks to a…
We give theorems about asymptotic normality of general additive functionals on patricia tries in an i.i.d. setting, derived from results on tries by Janson (2022). These theorems are applied to show asymptotic normality of the distribution…
We study a branching-process random iterated function system (RIFS) defined by a recursive replacement of leaves by finite subtrees at strictly smaller contraction scales. This construction yields a tree-valued, infinite-depth random…
We derive the exact partition function for a discrete model of random trees embedded in a one-dimensional space. These trees have vertices labeled by integers representing their position in the target space, with the SOS constraint that…
Rooted plane trees are reduced by four different operations on the fringe. The number of surviving nodes after reducing the tree repeatedly for a fixed number of times is asymptotically analyzed. The four different operations include…
In a supercritical branching particle system, the trimmed tree consists of those particles which have descendants at all times. We develop this concept in the superprocess setting. For a class of continuous superprocesses with Feller…
Subtraction-free computational complexity is the version of arithmetic circuit complexity that allows only three operations: addition, multiplication, and division. We use cluster transformations to design efficient subtraction-free…
In many domains, data measurements can naturally be associated with the leaves of a tree, expressing the relationships among these measurements. For example, companies belong to industries, which in turn belong to ever coarser divisions…
Search trees are fundamental data structures in computer science. We study functionals on random search trees that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal…
In the critical beta-splitting model of a random $n$-leaf binary tree, leaf-sets are recursively split into subsets, and a set of $m$ leaves is split into subsets containing $i$ and $m-i$ leaves with probabilities proportional to…
Phylogenetic trees are leaf-labelled trees used to model the evolution of species. In practice it is not uncommon to obtain two topologically distinct trees for the same set of species, and this motivates the use of distance measures to…
Tracking the movement of tracer particles has long been a strategy for uncovering complex structures. Here, we study discrete-time random walks on finite Cayley trees to infer key parameters such as tree depth and geometric bias toward the…
We consider the hierarchic tree Random Energy Model with continuous branching and calculate the moments of the corresponding partition function. We establish the multifractal properties of those moments. We derive formulas for the normal…
Additive tree functionals allow to represent the cost of many divide-and-conquer algorithms. We give an invariance principle for such tree functionals for the Catalan model (random tree uniformly distributed among the full binary ordered…
We investigate the statistics of trees grown from some initial tree by attaching links to preexisting vertices, with attachment probabilities depending only on the valence of these vertices. We consider the asymptotic mass distribution that…
Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis. We introduce a new class of upper bounds on the log partition…