The Distribution of Patterns in Random Trees

Let $T\_n$ denote the set of unrooted labeled trees of size $n$ and let $T\_n$ be a particular (finite, unlabeled) tree. Assuming that every tree of $T\_n$ is equally likely, it is shown that the limiting distribution as $n$ goes to infinity of the number of occurrences of $M$ as an induced subtree is asymptotically normal with mean value and variance asymptotically equivalent to $\mu n$ and $\sigma^2n$, respectively, where the constants $\mu>0$ and $\sigma\ge 0$ are computable

Non-Contiguous Pattern Avoidance in Binary Trees

In this paper we consider the enumeration of binary trees avoiding non-contiguous binary tree patterns. We begin by computing closed formulas for the number of trees avoiding a single binary tree pattern with 4 or fewer leaves and compare these results to analogous work for contiguous tree patterns. Next, we give an explicit generating function that counts binary trees avoiding a single non-contiguous tree pattern according to number of leaves. In addition, we enumerate binary trees that simultaneously avoid more than one tree pattern. Finally, we explore connections between pattern-avoiding trees and pattern-avoiding permutations.Comment: 21 pages, 2 figures, 1 tabl

Spatial patterns in mesic savannas: the local facilitation limit and the role of demographic stochasticity

We propose a model equation for the dynamics of tree density in mesic savannas. It considers long-range competition among trees and the effect of fire acting as a local facilitation mechanism. Despite short-range facilitation is taken to the local-range limit, the standard full spectrum of spatial structures obtained in general vegetation models is recovered. Long-range competition is thus the key ingredient for the development of patterns. The long time coexistence between trees and grass, and how fires affect the survival of trees as well as the maintenance of the patterns is studied. The influence of demographic noise is analyzed. The stochastic system, under the parameter constraints typical of mesic savannas, shows irregular patterns characteristics of realistic situations. The coexistence of trees and grass still remains at reasonable noise intensities.Comment: 12 pages, 7 figure

Pattern avoidance in labelled trees

We discuss a new notion of pattern avoidance motivated by the operad theory: pattern avoidance in planar labelled trees. It is a generalisation of various types of consecutive pattern avoidance studied before: consecutive patterns in words, permutations, coloured permutations etc. The notion of Wilf equivalence for patterns in permutations admits a straightforward generalisation for (sets of) tree patterns; we describe classes for trees with small numbers of leaves, and give several bijections between trees avoiding pattern sets from the same class. We also explain a few general results for tree pattern avoidance, both for the exact and the asymptotic enumeration.Comment: 27 pages, corrected various misprints, added an appendix explaining the operadic contex

Biomass partitioning and growth efficiency in four naturally regenerated forest tree species

Current forest growth models and yield tables are almost exclusively based on data from mature trees, reducing their applicability to young and developing stands. To address this gap, young European beech, sessile oak, Scots pine and Norway spruce trees approximately 0 to 10 years old were destructively sampled in a range of naturally regenerated forest stands in Central Europe. Diameter at base and height were first measured in situ for up to 175 individuals per species. Subsequently, the trees were excavated and dry biomass of foliage, branches, stems and roots was measured. Allometric relations were then used to calculate biomass allocation coefficients (BAC) and growth efficiency (GE) patterns in young trees. We found large differences in BAC and GE between broadleaves and conifers, but also between species within these categories. Both BAC and GE are strongly age-specific in young trees, their rapidly changing values reflecting different growth strategies in the earliest stages of growth. We show that linear relationships describing biomass allocation in older trees are not applicable in young trees. To accurately predict forest biomass and carbon stocks, forest growth models need to include species and age specific parameters of biomass allocation patterns

The influence of tree age and microhabitat quality on the occurrence of crustose lichens associated with old oaks

Questions: How do tree age, microhabitat characteristics and epiphytic competitors affect the occurrence of crustose lichens associated with old oaks? How do microhabitat characteristics and microclimate affect the cover of competitors (bryophytes and macrolichens)? How do microhabitat characteristics cor¬relate with microclimatic variables? Location: Southeast Sweden. Methods: Eight crustose lichen species were surveyed on 165 Quercus robur trees, 17-478 years old, at three study sites. The occurrence patterns of these species were examined at two spatial scales: among trees and within trees. Occurrence patterns within trees were examined in 10 cm × 10 cm plots at all four cardinal aspects at different heights from 0.5 to 4.5 m above the ground. Results: At the tree level, age-related factors were the most important predictors of species occurrence. All species were more frequent on trees > 100 years than on younger trees. At the plot level, the frequency of occurrence increased with increasing bark crevice depth. The frequencies of all study species de¬creased with increasing cover of bryophytes. Bryophytes were in turn more frequent in plots that were exposed to rainwater and showed a low evaporation rate. Patches most exposed to rainwater were directed upwards, and the lowest evaporation rates occurred on the northern side of the trunks. Conclusions: For many crustose lichens the association with old oak trees seems at least partly to depend on their preference for the deep bark crevices that only occur on old trees. Trees represent epiphyte habitat patches that differ in size due to within-tree variability in habitat quality, such as bark crevice depth and microclimate. This study shows that variability at a finer scale, within habitat patches, contribute to explain species occurrence patterns at habitat patch level

Effects of memory on the shapes of simple outbreak trees

Genomic tools, including phylogenetic trees derived from sequence data, are increasingly used to understand outbreaks of infectious diseases. One challenge is to link phylogenetic trees to patterns of transmission. Particularly in bacteria that cause chronic infections, this inference is affected by variable infectious periods and infectivity over time. It is known that non-exponential infectious periods can have substantial effects on pathogens’ transmission dynamics. Here we ask how this non-Markovian nature of an outbreak process affects the branching trees describing that process, with particular focus on tree shapes. We simulate Crump-Mode-Jagers branching processes and compare different patterns of infectivity over time. We find that memory (non-Markovian-ness) in the process can have a pronounced effect on the shapes of the outbreak’s branching pattern. However, memory also has a pronounced effect on the sizes of the trees, even when the duration of the simulation is fixed. When the sizes of the trees are constrained to a constant value, memory in our processes has little direct effect on tree shapes, but can bias inference of the birth rate from trees. We compare simulated branching trees to phylogenetic trees from an outbreak of tuberculosis in Canada, and discuss the relevance of memory to this dataset

Use of sonic tomography to detect and quantify wood decay in living trees.

Premise of the studyField methodology and image analysis protocols using acoustic tomography were developed and evaluated as a tool to estimate the amount of internal decay and damage of living trees, with special attention to tropical rainforest trees with irregular trunk shapes.Methods and resultsLiving trunks of a diversity of tree species in tropical rainforests in the Republic of Panama were scanned using an Argus Electronic PiCUS 3 Sonic Tomograph and evaluated for the amount and patterns of internal decay. A protocol using ImageJ analysis software was used to quantify the proportions of intact and compromised wood. The protocols provide replicable estimates of internal decay and cavities for trees of varying shapes, wood density, and bark thickness.ConclusionsSonic tomography, coupled with image analysis, provides an efficient, noninvasive approach to evaluate decay patterns and structural integrity of even irregularly shaped living trees

Biodiversity shapes tree species aggregations in tropical forests

Spatial patterns of conspecific trees are considered as the consequences of biological interactions and environmental influences. They also reflect species interactions in plant communities. However, biological attributes are often neglected while deliberating the factors shaping species distributions. As rising attentions are paid to spatial patterns of tropical forest trees, we noticed that seven Center of Tropical Forest Sites and four Forest Dynamic Plots in Asia and America have presented analogously high proportions of species with aggregated conspecific individuals coincidently. This phenomenon is distinctive and repudiates fundamental ecology hypotheses which suggested dispersed distributions of conspecific tropical trees due to intensive density and natural enemy pressures in tropical forests. We believe that similar aggregation patterns shared by these tropical forests implies the existence of structuring forces in biogeographical scale instead of habitat heterogeneity in local community scales as scientists have considered. To approach the factors contributing to this cross-continent spatial pattern of trees, we obtained and reviewed ecosystem attributes, including topography, temperature, precipitation, biodiversity, density, and biomass, of these forests. Here we show that the proportions of aggregated species are actually constants independent of any ecosystem attributes regardless the nature of these tropical forests. However, local biodiversity are the major factor determining the number of aggregated species and the aggregation of large individuals of these forests. Aggregation of large trees declines along rising biodiversity, while the numbers of aggregated species increase permanently along lifting biodiversity. We propose a possible equilibrium and saturated status of the tropical forests in accommodating aggregated species. Furthermore, the tight correlations of biodiversity and species aggregation strongly imply the importance of overlooked biological interactions in shaping the spatial patterns in the tropical forests