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.

An experiment was conducted in 2017 and 2018 to determine the sensitivity of driftable rates of 2,4-D and dicamba with or without glyphosate on common ornamental, fruit, and nut species. Three driftable rates corresponding to ½, 1/20th, and 1/200th of the manufacturer’s labeled rate (1 × rate) of 2,4-D (1.09 kg ae ha−1), 2,4-D plus glyphosate (1.09 kg ae ha−1 plus 1.10 kg ae ha−1), dicamba (0.56 kg ae ha−1), and dicamba plus glyphosate (0.56 kg ae ha−1 plus 1.10 kg ae ha−1) were applied to apple, crabapple, dogwood, American elderberry, American elm, grapevine, hydrangea, red maple, pin oak, peach, pecan, eastern redbud, rose, red raspberry, strawberry, sweetgum, nannyberry viburnum, and black walnut plants. Visible estimates of injury were recorded 28 and 56 days after treatment (DAT). Plant measurements included leaf malformation, tree trunk growth, and shoot length. Across all species, the ½ × rate of 2,4-D plus glyphosate resulted in 61% injury 28 DAT, whereas the ½ × rate of dicamba plus glyphosate resulted in 51% injury. Across plant species and herbicides, ½ ×, 1/20 ×, and 1/200 × rates caused injury ranging from 3% to 100%, 0% to 66%, and 0% to 19%, respectively. Hydrangea was the least sensitive species; grapevine was most sensitive. Changes in plant measurements were dependent on the species and herbicide applied. Treatments at the ½ × or 1/20 × rate resulted in shoot length, leaf malformation, and trunk tree diameter differences for 11, 10, and 7 species, respectively, compared with nontreated plants. Collectively, the measurements and visual injury assessments indicated apple, red maple, peach, and pin oak were more sensitive to treatments containing dicamba, whereas black walnut, grapevine, and American elm were more sensitive to 2,4-D. Although the 1/200 × rates of 2,4-D and dicamba did not result in changes to plant measurements, obvious injury symptoms were observed, which could render these plants unsalable.

The classical Halpern–Läuchli theorem states that for any finite coloring of a finite product of finitely branching perfect trees of height ω, there exist strong subtrees sharing the same level set such that tuples in the product of the strong subtrees consisting of elements lying on the same level get the same color. Relative to large cardinals, we establish the consistency of a tail cone version of the Halpern–Läuchli theorem at a large cardinal (see Theorem 3.1), which, roughly speaking, deals with many colorings simultaneously and diagonally. Among other applications, we generalize a polarized partition relation on rational numbers due to Laver and Galvin to one on linear orders of larger saturation.

People's relationships with trees reflect the landscape histories associated with distinctive forms of political and religious authority and the moral imaginings of people in Nangodi and its environs in the Upper East Region of Ghana. Their memories of particular trees serve as historical evidence of overlapping yet specific forms of political authority exercised by chiefs, earth priests, past colonial officers, and present-day Ghanaian government officials. In Nangodi, individual family ancestral tree shrines, clan tree cemeteries, and sacred groves associated with earth priests and chiefs coexist with the Red Volta West Forest Reserve and with a succession of tree-related development initiatives. While these relationships are often considered as separate claims to political authority, spiritual power, or scientific knowledge, this paper argues that these relationships of people and of trees are better conceptualized as historical accumulations that represent intersecting and contested forms of authority and political rule continuing into the present. Indigenous tree species such as ebony are associated with sacred groves controlled by chiefs, silk-cotton trees with earth priests’ cemeteries, and baobab trees with particular families coexist with foreign teak trees associated with colonial forestry. This situation suggests how institutions of governance as well as the actions of individuals have environmental consequences. A consideration of historical memories of people and trees in places such as Nangodi enables a rethinking of political and environmental dichotomies, and complicates the social dynamics of` the preservation and destruction of trees and forests around the world.

We define a growing model of random graphs. Given a sequence of non-negative integers {dn}n=0∞ with the property that di≤i, we construct a random graph on countably infinitely many vertices v0, v1… by the following process: vertex vi is connected to a subset of {v0, …, vi−1} of cardinality di chosen uniformly at random. We study the resulting probability space. In particular, we give a new characterization of random graphs, and we also give probabilistic methods for constructing infinite random trees.

Long-term forest dynamics plots in the tropics tend to be situated on stable terrain. This study investigated forest dynamics on the north coast of New Guinea where active subduction zones are uplifting lowland basins and exposing relatively young sediments to rapid weathering. We examined forest dynamics in relation to disturbance history, topography and soil nutrients based on partial re-census of the 50-ha Wanang Forest Dynamics Plot in Papua New Guinea. The plot is relatively high in cations and phosphorus but low in nitrogen. Soil nutrients and topography accounted for 29% of variation in species composition but only 4% of variation in basal area. There were few areas of high biomass and most of the forest was comprised of small-diameter stems. Approximately 18% of the forest was less than 30 y old and the annual tree mortality rate of nearly 4% was higher than in other tropical forests in South-East Asia and the neotropics. These results support the reputation of New Guinea's forests as highly dynamic, with frequent natural disturbance. Empirical documentation of this hypothesis expands our understanding of tropical forest dynamics and suggests that geomorphology might be incorporated in models of global carbon storage especially in regions of unstable terrain.