We show that for every $\eta \gt 0$ every sufficiently large $n$-vertex oriented graph $D$ of minimum semidegree exceeding $(1+\eta )\frac k2$ contains every balanced antidirected tree with $k$ edges and bounded maximum degree, if $k\ge \eta n$. In particular, this asymptotically confirms a conjecture of the first author for long antidirected paths and dense digraphs.

Further, we show that in the same setting, $D$ contains every $k$-edge antidirected subdivision of a sufficiently small complete graph, if the paths of the subdivision that have length $1$ or $2$ span a forest. As a special case, we can find all antidirected cycles of length at most $k$.

Finally, we address a conjecture of Addario-Berry, Havet, Linhares Sales, Reed, and Thomassé for antidirected trees in digraphs. We show that this conjecture is asymptotically true in $n$-vertex oriented graphs for all balanced antidirected trees of bounded maximum degree and of size linear in $n$.

In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by $\alpha $-CLI and L-$\alpha $-CLI where $\alpha $ is a countable ordinal. We establish three results:

1. (1) G is $0$-CLI iff $G=\{1_G\}$;

2. (2) G is $1$-CLI iff G admits a compatible complete two-sided invariant metric; and

3. (3) G is L-$\alpha $-CLI iff G is locally $\alpha $-CLI, i.e., G contains an open subgroup that is $\alpha $-CLI.

Subsequently, we show this hierarchy is proper by constructing non-archimedean CLI Polish groups $G_\alpha $ and $H_\alpha $ for $\alpha <\omega _1$, such that:

1. (1) $H_\alpha $ is $\alpha $-CLI but not L-$\beta $-CLI for $\beta <\alpha $; and

2. (2) $G_\alpha $ is $(\alpha +1)$-CLI but not L-$\alpha $-CLI.

We study the possible structures which can be carried by sets which have no countable subset, but which fail to be ‘surjectively Dedekind finite’, in two possible senses, that there is surjection to $\omega $, or alternatively, that there is a surjection to a proper superset.

Let T be the regular tree in which every vertex has exactly $d\ge 3$ neighbours. Run a branching random walk on T, in which at each time step every particle gives birth to a random number of children with mean d and finite variance, and each of these children moves independently to a uniformly chosen neighbour of its parent. We show that, starting with one particle at some vertex 0 and conditionally on survival of the process, the time it takes for every vertex within distance r of 0 to be hit by a particle of the branching random walk is $r + ({2}/{\log(3/2)})\log\log r + {\mathrm{o}}(\log\log r)$ .

We study the first-order theories of some natural and important classes of coloured trees, including the four classes of trees whose paths have the order type respectively of the natural numbers, the integers, the rationals, and the reals. We develop a technique for approximating a tree as a suitably coloured linear order. We then present the first-order theories of certain classes of coloured linear orders and use them, along with the approximating technique, to establish complete axiomatisations of the four classes of trees mentioned above.

Vatica cauliflora P.S. Ashton (Dipterocarpaceae) is a threatened tree species endemic to Kapuas Hulu District, West Kalimantan Province, Indonesia. The species is only known from the type specimen collected in 1953. After this first collection, there was no record confirming the presence of this tree in its natural habitat. Our recent surveys in 2019 and 2020 located 179 individuals of the species in six unprotected locations. The population's size structure is dominated (62.6%) by young individuals within the 0–5 cm diameter class. Our surveys also showed that the habitat of V. cauliflora is degraded as a result of the negative effects of agriculture and logging. Assessment with the IUCN Red List criteria indicates that V. cauliflora should be categorized as Critically Endangered.

We show that the countable universal homogeneous meet-tree has a generic automorphism, but it does not have a generic pair of automorphisms.

In this paper, we mainly study a class of small deviation theorems for Markov chains indexed by an infinite tree with uniformly bounded degree in Markovian environment. Firstly, we give the definition of Markov chains indexed by a tree with uniformly bounded degree in random environment. Then, we introduce the some lemmas which are the basis of the results. Finally, a class of small deviation theorems for functionals of random fields on a tree with uniformly bounded degree in Markovian environment is established.

Acer yangbiense Y.S. Chen & Q.E. Yang (Aceraceae) is a threatened tree species endemic to China, formerly presumed to have declined to only five extant individuals, restricted to Yangbi County, Yunnan Province. Our surveys in 2016, however, located 577 individuals in 12 localities, but only three localities (with a total of 62 individuals) are protected. Nine localities are on private forest land. The population's size structure is an inverse J-curve, but there is a scarcity of trees of the smallest size class and of seedlings. Our surveys also showed that the habitat of A. yangbiense is degraded as a result of the negative effects of agriculture, logging and wood harvesting. Assessment with the IUCN Red List categories and criteria indicates that A. yangbiense should be recategorized from Critically Endangered to Endangered.

Let $\unicode[STIX]{x1D6E4}\leqslant \text{Aut}(T_{d_{1}})\times \text{Aut}(T_{d_{2}})$ be a group acting freely and transitively on the product of two regular trees of degree $d_{1}$ and $d_{2}$. We develop an algorithm that computes the closure of the projection of $\unicode[STIX]{x1D6E4}$ on $\text{Aut}(T_{d_{t}})$ under the hypothesis that $d_{t}\geqslant 6$ is even and that the local action of $\unicode[STIX]{x1D6E4}$ on $T_{d_{t}}$ contains $\text{Alt}(d_{t})$. We show that if $\unicode[STIX]{x1D6E4}$ is torsion-free and $d_{1}=d_{2}=6$, exactly seven closed subgroups of $\text{Aut}(T_{6})$ arise in this way. We also construct two new infinite families of virtually simple lattices in $\text{Aut}(T_{6})\times \text{Aut}(T_{4n})$ and in $\text{Aut}(T_{2n})\times \text{Aut}(T_{2n+1})$, respectively, for all $n\geqslant 2$. In particular, we provide an explicit presentation of a torsion-free infinite simple group on 5 generators and 10 relations, that splits as an amalgamated free product of two copies of $F_{3}$ over $F_{11}$. We include information arising from computer-assisted exhaustive searches of lattices in products of trees of small degrees. In an appendix by Pierre-Emmanuel Caprace, some of our results are used to show that abstract and relative commensurator groups of free groups are almost simple, providing partial answers to questions of Lubotzky and Lubotzky–Mozes–Zimmer.

In this paper, we extend the strong laws of large numbers and entropy ergodic theorem for partial sums for tree-indexed nonhomogeneous Markov chains fields to delayed versions of nonhomogeneous Markov chains fields indexed by a homogeneous tree. At first we study a generalized strong limit theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Then we prove the generalized strong laws of large numbers and the generalized asymptotic equipartition property for delayed sums of finite nonhomogeneous Markov chains indexed by a homogeneous tree. As corollaries, we can get the similar results of some current literatures. In this paper, the problem settings may not allow to use Doob's martingale convergence theorem, and we overcome this difficulty by using Borel–Cantelli Lemma so that our proof technique also has some new elements compared with the reference Yang and Ye (2007).

Ruminant-based food production faces currently multiple challenges such as environmental emissions, climate change and accelerating food–feed–fuel competition for arable land. Therefore, more sustainable feed production is needed together with the exploitation of novel resources. In addition to numerous food industry (milling, sugar, starch, alcohol or plant oil) side streams already in use, new ones such as vegetable and fruit residues are explored, but their conservation is challenging and production often seasonal. In the temperate zones, lipid-rich camelina (Camelina sativa) expeller as an example of oilseed by-products has potential to enrich ruminant milk and meat fat with bioactive trans-11 18:1 and cis-9,trans-11 18:2 fatty acids and mitigate methane emissions. Regardless of the lower methionine content of alternative grain legume protein relative to soya bean meal (Glycine max), the lactation performance or the growth of ruminants fed faba beans (Vicia faba), peas (Pisum sativum) and lupins (Lupinus sp.) are comparable. Wood is the most abundant carbohydrate worldwide, but agroforestry approaches in ruminant nutrition are not common in the temperate areas. Untreated wood is poorly utilised by ruminants because of linkages between cellulose and lignin, but the utilisability can be improved by various processing methods. In the tropics, the leaves of fodder trees and shrubs (e.g. cassava (Manihot esculenta), Leucaena sp., Flemingia sp.) are good protein supplements for ruminants. A food–feed production system integrates the leaves and the by-products of on-farm food production to grass production in ruminant feeding. It can improve animal performance sustainably at smallholder farms. For larger-scale animal production, detoxified jatropha (Jatropha sp.) meal is a noteworthy alternative protein source. Globally, the advantages of single-cell protein (bacteria, yeast, fungi, microalgae) and aquatic biomass (seaweed, duckweed) over land crops are the independence of production from arable land and weather. The chemical composition of these feeds varies widely depending on the species and growth conditions. Microalgae have shown good potential both as lipid (e.g. Schizochytrium sp.) and protein supplements (e.g. Spirulina platensis) for ruminants. To conclude, various novel or underexploited feeds have potential to replace or supplement the traditional crops in ruminant rations. In the short-term, N-fixing grain legumes, oilseeds such as camelina and increased use of food and/or fuel industry by-products have the greatest potential to replace or supplement the traditional crops especially in the temperate zones. In the long-term, microalgae and duckweed of high-yield potential as well as wood industry by-products may become economically competitive feed options worldwide.

For two given graphs $G_{1}$ and $G_{2}$, the planar Ramsey number $PR(G_{1},G_{2})$ is the smallest integer $N$ such that every planar graph $G$ on $N$ vertices either contains $G_{1}$, or its complement contains $G_{2}$. Let $C_{4}$ be a quadrilateral, $T_{n}$ a tree of order $n\geq 3$ with maximum degree $k$, and $K_{1,k}$ a star of order $k+1$. We show that $PR(C_{4},T_{n})=\max \{n+1,PR(C_{4},K_{1,k})\}$. Combining this with a result of Chen et al. [‘All quadrilateral-wheel planar Ramsey numbers’, Graphs Combin.33 (2017), 335–346] yields exact values of all the quadrilateral-tree planar Ramsey numbers.

In southern African savannas, geoxylic suffrutices or ‘underground trees’ attain only a hundredth to a tenth the height of normal trees, but other traits have received little attention. Geoxylic suffrutices and congeneric trees were compared for minimum and maximum values of seven morphological traits. Thirty-six geoxyle-tree pairs co-occurring in Katanga (Democratic Republic of the Congo) were compared, based on data from standard floras. The tree/geoxyle ratio ranged from 0.92 to 1.67 and was greater than 1 in 12 of 14 trait comparisons. However, the difference was significant in only five comparisons. Reproductive traits generally did not differ. The maximal value of leaf traits (lamina length, lamina width, petiole length) was 33–67% greater in trees. The morphological traits of geoxyles are not much altered compared with their tree counterparts, especially for reproductive traits. For vegetative traits, geoxyles express a restricted part of the phenetic space of trees, being unable to attain trait values as high as those of their tree congeners. However, unlike bonsais or alpine dwarfs, the leaves of geoxyles are not much smaller compared with normal trees.

Let ${{G}_{1}},\,{{G}_{2}},\,.\,.\,.\,,\,{{G}_{t}}$ be arbitrary graphs. The Ramsey number $R\left( {{G}_{1}},\,{{G}_{2}},\,.\,.\,.,{{G}_{t}} \right)$ is the smallest positive integer $n$ such that if the edges of the complete graph ${{K}_{n}}$ are partitioned into $t$ disjoint color classes giving $t$ graphs ${{H}_{1}},\,{{H}_{2}},\,.\,.\,.\,,\,{{H}_{t}}$ , then at least one ${{H}_{i}}$ has a subgraph isomorphic to ${{G}_{i}}$ . In this paper, we provide the exact value of the $R({{T}_{n}},\,{{W}_{m}})$ for odd $m,\,n\,\ge \,m-1$ , where ${{T}_{n}}$ is either a caterpillar, a tree with diameter at most four, or a tree with a vertex adjacent to at least $\left\lceil \frac{n}{2} \right\rceil \,-\,2$ leaves, and ${{W}_{n}}$ is the wheel on $n\,+\,1$ vertices. Also, we determine $R\left( {{C}_{n}},\,{{W}_{m}} \right)$ for even integers $n$ and $m,\,n\,\ge \,m\,+\,500$ , which improves a result of Shi and confirms a conjecture of Surahmat et al. In addition, the multicolor Ramsey number of trees versus an odd wheel is discussed in this paper.