A least-squares algorithm for fitting ultrametric and path length or additive trees to two-way, two-mode proximity data is presented. The algorithm utilizes a penalty function to enforce the ultrametric inequality generalized for asymmetric, and generally rectangular (rather than square) proximity matrices in estimating an ultrametric tree. This stage is used in an alternating least-squares fashion with closed-form formulas for estimating path length constants for deriving path length trees. The algorithm is evaluated via two Monte Carlo studies. Examples of fitting ultrametric and path length trees are presented.

The management and mismanagement of Roman groves was a serious matter, and intentional and unintentional violations of these spaces could be severely punished. In spite of this, groves remained loosely defined by Romans and their boundaries were commonly misunderstood, a confusion that has continued into modern scholarship, where groves are understood as either a clearing in a wood or a dark space lit by artificial lighting. This article takes up this discussion, and explores the nature of an ancient grove as a well-attested space under forest management that influences later conversations on the nature of wooded spaces in more recent periods.

Collaborative autoethnography can function as a means of reclaiming certain African realities that have been co-opted by colonial epistemes and language. This can be significant in very concrete ways: northern Uganda is suffering a catastrophic loss of tree cover, much of which is taking place on the collective family landholdings that academia and the development sector have categorized as “customary land.” A collaboration by ten members of such landholding families, known as the Acholi Land Lab, explores what “customary ownership” means to them and their relatives, with a view to understanding what may be involved in promoting sustainable domestic use of natural resources, including trees.

We study two models of discrete height functions, that is, models of random integer-valued functions on the vertices of a tree. First, we consider the random homomorphism model, in which neighbours must have a height difference of exactly one. The local law is uniform by definition. We prove that the height variance of this model is bounded, uniformly over all boundary conditions (both in terms of location and boundary heights). This implies a strong notion of localisation, uniformly over all extremal Gibbs measures of the system. For the second model, we consider directed trees, in which each vertex has exactly one parent and at least two children. We consider the locally uniform law on height functions which are monotone, that is, such that the height of the parent vertex is always at least the height of the child vertex. We provide a complete classification of all extremal gradient Gibbs measures, and describe exactly the localisation-delocalisation transition for this model. Typical extremal gradient Gibbs measures are localised also in this case. Localisation in both models is consistent with the observation that the Gaussian free field is localised on trees, which is an immediate consequence of transience of the random walk.

Elementary first-order theories of trees allowing at most, exactly $\mathrm{m}$, and any finite number of immediate descendants are introduced and proved mutually interpretable among themselves and with Robinson arithmetic, Adjunctive Set Theory with Extensionality and other well-known weak theories of numbers, sets, and strings.

Found only in a restricted area of north-west Australia, the Australian boab (Adansonia gregorii) is recognisable by its massive, bottle-shaped trunk, and is an economically important species for Indigenous Australians, with the pith, seeds and young roots all eaten. Many of these trees are also culturally significant and are sometimes carved with images and symbols. The authors discuss the history of research into carved boabs in Australia, and present a recent survey to locate and record these trees in the remote Tanami Desert. Their results provide insight into the archaeological and anthropological significance of dendroglyphs in this region and add to a growing corpus of information on culturally modified trees globally.

Root metrics and plant height for 256 excavated saplings and small trees of 27 species, including sown plants, were used to describe belowground structure and assess factors that influence shoot growth in a tropical dry forest (TDF) in Zambia. Models were developed to (i) estimate taproot depth from incomplete excavations and (ii) coarse lateral root biomass from proximal diameter data. The majority of the species studied are slow-growing and had a median height of <200 cm at the age of 16 years. Root development advanced sequentially from taproot elongation to thickening to coarse lateral root development. Shrubs in shallow soil had short taproots with a lower wood density. Plant age explained <10% of the variance in shoot height. Root variables explained the majority of the variance in shoot height. More research is needed to improve our knowledge about how belowground structures influence shoot growth and tree recruitment in TDFs of southern Africa.

Assuming $\mathrm{PFA}$ , we shall use internally club $\omega _1$ -guessing models as side conditions to show that for every tree T of height $\omega _2$ without cofinal branches, there is a proper and $\aleph _2$ -preserving forcing notion with finite conditions which specialises T. Moreover, the forcing has the $\omega _1$ -approximation property.

We make comments on some problems Erdős and Hajnal posed in their famous problem list. Let X be a graph on $\omega _1$ with the property that every uncountable set A of vertices contains a finite set s such that each element of $A-s$ is joined to one of the elements of s. Does then X contain an uncountable clique? (Problem 69) We prove that both the statement and its negation are consistent. Do there exist circuitfree graphs $\{X_n:n<\omega \}$ on $\omega _1$ such that if $A\in [\omega _1]^{\aleph _1}$ , then $\{n<\omega :X_n\cap [A]^2=\emptyset \}$ is finite? (Problem 61) We show that the answer is yes under CH, and no under Martin’s axiom. Does there exist $F:[\omega _1]^2\to 3$ with all three colors appearing in every uncountable set, and with no triangle of three colors. (Problem 68) We give a different proof of Todorcevic’ theorem that the existence of a $\kappa $ -Suslin tree gives $F:[\kappa ]^2\to \kappa $ establishing $\kappa \not \to [\kappa ]^2_{\kappa }$ with no three-colored triangles. This statement in turn implies the existence of a $\kappa $ -Aronszajn tree.

What is the maximum number of copies of a fixed forest T in an n-vertex graph in a graph class $\mathcal {G}$ as $n\to \infty $ ? We answer this question for a variety of sparse graph classes $\mathcal {G}$ . In particular, we show that the answer is $\Theta (n^{\alpha _{d}(T)})$ where $\alpha _{d}(T)$ is the size of the largest stable set in the subforest of T induced by the vertices of degree at most d, for some integer d that depends on $\mathcal {G}$ . For example, when $\mathcal {G}$ is the class of k-degenerate graphs then $d=k$ ; when $\mathcal {G}$ is the class of graphs containing no $K_{s,t}$ -minor ( $t\geqslant s$ ) then $d=s-1$ ; and when $\mathcal {G}$ is the class of k-planar graphs then $d=2$ . All these results are in fact consequences of a single lemma in terms of a finite set of excluded subgraphs.

The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees.