An easy, convenient statistical method to analyze bark beetle populations

A non-invasive method to estimate the number of Trypodendron lineatum holes on dead standing pines (Pinus sylvestris L.) was developed using linear and nonlinear estimations. A classical linear regression model was first used to analyze the relationship between the number of holes caused by T. lineatum on selected stem units and the total number of holes on an entire dead stem of P. sylvestris. Then, to obtain a better fit of the regression function to the data for the stem unit selected in the first step, piecewise linear regression (PLR) was used. Last, in an area used to evaluate wood decomposition (method validation), the total and mean numbers of T. lineatum holes were estimated for single dead trees and for a sample (n = 8 dead trees). Data were collected in 2009 (data set D1), in 2010–2014 (data set D2) and in 2020 (data set D3) in forests containing P. sylvestris located within Suchedniów–Oblęgorek Landscape Park, Poland. A model was constructed with three linear equations. An evaluation of model accuracy showed that it was highly effective regardless of the density of T. lineatum holes and sample size. The method enables the evaluation of the biological role of this species in the decomposition of dead standing wood of P. sylvestris in strictly protected areas.


Introduction
Dead wood is a key factor in the maintenance of forest biodiversity (e.g., Harmon et al. 1986;Stockland et al. 2012;Ulyshen and Šobotník 2018), and research on the rates of decomposition of dead wood has been increasing rapidly (e.g., Duffy et al. 2017;Eichenberg et al. 2017;Kahl et al. 2017;Pietsch et al. 2019).The contributions of the main decomposers-fungi, bacteria, and invertebrates in the breakdown of wood, however, is not fully understood (e.g., Pietsch et al. 2019;Wang et al. 2022).Some studies indicate that they make a large contribution, around 10-20%, to the mass loss of the annual input of dead wood (e.g., Ulyshen et al. 2016), but other studies indicate a more limited impact (Ulyshen and Wagner 2013;Eichenberg et al. 2017).
Among invertebrate wood decomposers, an important group are saproxylic insects (Stockland et al. 2012), which include Trypodendron lineatum Olivier 1795 (Coleoptera: Curculionidae: Scolytinae), a widely distributed species in the Palearctic region and North America (Wood 1982).It is a polyphagous and found in every stand containing conifers (Chapman 1966;Borden 1988;Park and Reid 2007).It favors shady sites and spruce and pine species as its host (Oranen 2013), but it also attacks Douglas-fir (Pseudotsuga menziesii), larch (Tamarix sp.), western hemlock (Tsuga heterophylla)and western red cedar (Thuja plicata) (Oranen 2013).T. lineatum is also a common pest in Poland (Szujecki 1987) where it colonizes dying and dead standing trees, windthrows, stumps and debarked timber (Wood 1982;Borden 1988).Logs stored at forest sites or millyards with a suitable moisture content and stumps of 1-to 2-year-old trees provide a breeding ground for T. lineatum (Lindelöw et al. 1992;Orbay et al. 1994;Park and Reid 2007).
By colonizing greatly weakened but still-living trees and initiating their decomposition, T. lineatum acts as a pioneer species (Wood 1982;Lindgren 1990) in two ways.The first is associated with physical damage to the wood.The species is monogamous, and the female bores out an entrance gallery in the wood, about 5 cm long, perpendicular to the annual rings.From the end of the entrance gallery, there are usually two maternal galleries branching out, usually running along an annual ring.The entrance holes are used as exit holes (Szujecki 1987;Beaver 1989).This damage changes the microclimatic conditions in the wood (better ventilation, higher moisture content), which favors faster decomposition (Stockland et al. 2012) and is particularly important for dead standing trees, since in contrast to fallen trees, they are not physically fragmented, which may occur when a tree falls, for example.The second method of decomposition relates to the introduction of fungal spores, mainly from the species Phialophoropsis ferruginea Math.-Käärik, into the wood by females as they bore galleries into the wood (Mayers et al. 2015).The spores can serve as food for the larvae and the eclosed adult beetles, and the hyphae that are produced can initiate wood decomposition, facilitating colonization by later species (Stockland et al. 2012;Runnel and Löhmus 2017).T. lineatum, like as all bark beetles, may be a vector for virulent or moderately virulent fungi (Jankowiak et al. 2017;Lehenberger et al. 2018) can decompose wood.Although the fungal communities inhabiting different woody niches change during succession and in association with the species of beetles (Persson et al. 2009;Oranen 2013;Lehenberger et al. 2019;Pan et al. 2020).
This manner of colonization and the reproductive strategy of T. lineatum imply that the rate of the initial phase of wood decomposition will be directly proportional to its population density on the colonized stem.Hence, knowledge of the size of its population is fundamental to determine its biological role in decomposition.Despite the fact that many risk models have been developed for bark beetles using classification and regression tree statistical techniques based on the population density of an endemic pest (Elkin and Reid 2010;Bretfeld et al. 2021), little work has been done to develop a statistical method to analyze populations of saproxylic insects such as T. lineatum.
There are many approaches to developing risk and threat-rate models to predict bark beetle infestation and tree mortality (Zolubas et al. 2009;Bretfeld et al. 2021).Traditional approaches are based on stand characteristics (e.g., susceptibility), physiological predictors (e.g., water, nutrients, phenols, resin), and intensive sampling of the pest in representative areas and have been tested and applied to aggressive bark beetles that are forest pests.Methods to estimate the population density of pests such as T. lineatum are also needed (Lieutier et al. 2015).
The most important part of estimating the density of T. lineatum is determining the total number of holes created by the insects on dead pines.It requires great precision and involves counting all the T. lineatum holes on every analyzed tree.So it is also highly labor-intensive, requiring the felling of dead pines and careful debarking of entire stems.The process also greatly interferes with the hygrothermal conditions in the stem.In general, the rate of decomposition is thought to be related to the type of material and the environmental conditions, especially temperature and humidity (Harmon et al. 1986;Weedon et al. 2009).
Thus, in developing a method to estimate the total number of T. lineatum holes on standing trees for the study of the rate of wood decomposition in forest ecosystems, two conditions must be met.(1) The method must be noninvasive; the tree cannot be felled, and holes should be counted only when the bark has naturally fallen from the stem.Dead standing trees must be preserved as critical niches for various groups of organisms, for which they are often obligate habitats (e.g., Siitonen et al. 1999;Bobiec et al. 2005;Vehkaoja et al. 2016;Santaniello et al. 2017).( 2) Because counting all holes on an entire tree is practically not feasible (requiring a mountaineering cable or mechanical lift), a statistical method is needed that uses models to estimate the total number of holes on pine, based for example on relationships between the number of holes on selected stem units and the total number of T. lineatum holes on the stem.In this case, the units should be situated as close as possible to the base of the stem.If the stem units are at a height of up to 3 m or a maximum of 4 m, it will be technically feasible to count holes of T. lineatum using a light ladder approximately 2-3 m tall and without felling the tree.Methods of this type have been used to estimate population densities of the three main species of bark beetle on windthrows of Abies alba in Świętokrzyski National Park, Poland (e.g., Podlaski and Borkowski 2009).
The broad distribution of of T. lineatum, its high plasticity in terms of colonizing breeding material and its pioneer role in wood decomposition are evidence of its great ecological significance.A rapid, accurate method to estimate the population density of T. lineatum and satisfies the prescribed conditions would facilitate analyses of the biological role of T. lineatum in forest ecosystems and be especially valuable in strictly protected areas of high natural value.To develop and validate such a statistical method based on the holes the insects created in stems, I estimated the total number of holes on the stems of dead trees based on the relationship between the number of holes on selected stem units and the total number of T. lineatum holes on the stem, using a twostage statistical process.I then fitted a regression function to the data for the stem unit for which the correlation was the strongest.

Study area
Fieldwork was carried out in the Świętokrzyskie Mountains in central Poland (20°45′ E; 50°55′ N; 200-400 m above sea level).Meteorological data for 2009-2019 were obtained from the Forest Data Bank (2018).The annual mean temperature was 7.5 °C, and the annual mean precipitation was 650 mm.The growing season (days with daily mean temperature above 5 °C) lasts from 1-5 April to 24-30 October.The area has southwesterly and westerly prevailing winds.The data were collected in the Suchedniowsko-Oblęgorski Landscape Park in near-natural forests that are the remnants of the Świętokrzyska Primeval Forest (Fig. 1A).The area's high natural value is reflected by the creation of strict nature reserves within the park.Moreover, all selected stands are in areas of upland mixed forest type.The trees of Pinus sylvestris L. were aged 80 years and that species made up approximately 40% of the stand.

Data sets
Data were collected in 2009 (data set D1) in 2010-2014 (data set D2) and in 2020 (data set D3).
In April 2009, all the dead trees were numbered (a total of 42 dead trees) in an area of approximately 5 ha.Then, 30 trees were selected randomly for entomological analyses (data set D1), using a random number generator (Fig. 1B) and simple random sampling without replacement (SRSWOR) (Cochran 1977;Thompson 2012).The sample size was limited to 30 trees, in accordance with nature protection requirements relating to the number of dead trees in management forests (Instructions for Forest Protection in Poland 2012).All the dead standing trees had bark; however, after felling, some of the dead trees were left partly without bark, mostly on the thicker part of the stem.
The remaining dead trees (12 trees) were left to decompose naturally (data set D2).In September 2010-2014, they were monitored to determine the amount of time for the bark to fall from the stem at the thickest part of the dead tree.In 2014, the dead standing trees left to decompose naturally lacked bark on a section up to at least 4 m measured from the base of the stem.
Data set D3 was collected in September 2020, using a method that was used to inventory dead wood in Białowieża National Park, Poland (Bobiec et al. 2005).Here, an area 500 m × 500 m was marked, selected so that one side of the square was more than 2000 m from the boundary of the forest.The sample trees were selected by extending a measuring tape along two series of lines that had a total length of 2 km.Each series consisted of mutually perpendicular 50 m sections (a stairway layout), unning along both diagonals of the area (Fig. 1C).Data on the number and colonization of dead standing trees were collected along the marked lines, in strips 30 m wide (15 m on each side of the line).Of eight dead trees analyzed, five lacked bark on a section up to at least 4 m from the base of the stem.

Field data
In the first stage, on the dead trees analyzed, the length of the section of stem without bark was evaluated (data sets D1 and D3).After the trees were felled, for each sample tree, we measured (1) stem diameter at the thickest and thinnest part, (2) diameter of each meter-long stem unit measured at its thickest and thinnest ends, and (3) the length of each stem.The mean diameter of dead trees was 30.8 cm (± 5.3 SD) in 2009 and 25.6 cm (± 3.9) in 2020.The mean length of dead trees was 20.5 cm (± 1.6) in 2009 and 18.6 cm (± 1.3) in 2020.The age of the trees was determined for 2009 as 80 years.In 2020, the age of dead trees was not assessed.
Variables for the colonization of the stem by bark beetles, were estimated using a method of sectional entomological analysis (Szujecki 1987).After removal of the bark, the degree of colonization of the stem by major bark beetle species was determined.On successive meter-long stem units, from the thicker end, holes made by T. lineatum and egg galleries made by Tomicus minor and Tomicus piniperda were counted.Colonization density on each pine stem was calculated by (1) summing the number of galleries on all sections and (2) calculating the side surface area of the pine stems using the formulae given by Borkowski and Podlaski (2011).
The stem form of a coniferous tree can be expressed by Kunze's equation (Inoue 2006): where r is the stem radius, l is stem length from tree tip, and b and c are coefficients.The stem surface area s of the tree can be calculated as: where h is the length of the stem.

Statistical method to estimate number of T. lineatum holes in trees in study area
A model to estimate the total number of T. lineatum holes was constructed using data from the 2009 entomological analysis (data set D1), and its accuracy was evaluated using data from 2020 (data set D3).The described model construction scheme, whereby the model is evaluated using new, independent data (Adlard 1995;Soares et al. 1995;Kozak and Kozak 2003) 2031 An easy, convenient statistical method to analyze bark beetle populations 1 3 is common forestry practice, particularly when forecasting changes in forest resources (e.g., Peng 2000).

Prediction of total number of T. lineatum holes on dead, standing tree stems of P. sylvestris
Population N denotes the total number of holes of T. lineatum on the stems either from observed data (N o ) or from values predicted by the model (N p ).The total number of T. lineatum holes on the dead trees was described using methods based on linear and nonlinear regression.In the first stage, correlation and regression analyses were used to determine the relationship between the number of T. lineatum holes on selected meter-long stem units and the total number of T. lineatum holes on the stem using the following linear regression function: where N p is the total number of holes on a whole P. sylvestris stem, a 0k and a 1k are parameters of the model, and n k is the number of T. lineatum holes on the kth distinguished meterlong stem unit (k = 1, 2,…, 24).
The regression function parameters (a 0k , a 1k ), the coefficients of correlation (R k ) and determination ( R 2 k ), and the mean relative error of estimation (sw k ) were calculated for each stem unit.The parameters of the linear regression functions were estimated by the least squares method.The null hypotheses H 0 , that the coefficients a 0k are equal to zero, were also tested.Following the calculations, the best function was chosen, that is, yielded the highest correlation coefficient and lowest mean relative error of estimation.
In the second stage, to obtain a better fit of the regression function for the fourth stem unit, piecewise linear regression (PLR) with one breakpoint (model I, Eq. 4) and with two breakpoints of the regression line (model II, Eq. 5) was used (Seber and Wild 2003): (3) where N p is the total number of holes on the whole P. sylvestris stem; a 0 , a 1 , b 0 , b 1 , c 0 and c 1 are parameters of the model; n 4 is the number of T. lineatum holes on the fourth stem unit; and α 1 and α 2 are the breakpoints.
The fit of the constructed models was assessed based on the coefficient of determination (R 2 ) and the loss function for the sum of least squares.A quasi-Newton method of estimation was used to find the minimum of the loss function (Chong and Zak 2001).The usefulness of the models was evaluated based on observed and residual values in relation to the values predicted according to the equations.The residual values were checked for a normal distribution using the Shapiro-Wilk test (Sokal and Rohlf 2012).These analyses were performed using the program Statistica v. 10.0 for Windows (StatSoft, Tulsa, OK, USA).

Evaluation of model accuracy
Application of the method requires the determination, for each equation, of a class dependent on the number of holes in the fourth stem unit.The classes were defined based on the values of the regression line breakpoints.The following classes were defined for the particular linear equations: (1) Class 1, for the equation below the first breakpoint (< 150 holes); (2) Class 2, for the equation between the first and second breakpoints (151-395 holes); (3) Class 3, for the equation above the second breakpoint (> 395 holes).
From data set D3, the total number of holes N was calculated for each dead tree, by summing the numbers of holes for all meter-long stem units (N o ) and applying the developed method to obtain N p .The accuracy of the method was evaluated for individual dead trees and for all dead trees.
For individual trees, the relative error was computed from the formula For all dead trees in the study area, the mean total number of holes of T. lineatum on the entire stem, N , is an unbiased estimator for N (Thompson 2012).where N is the mean and and s 2 is the variance of the N-val- ues in the sample, and n is the sample size (the number of all P. sylvestris dead trees in the sample).
The significance of any differences between the mean observed number of holes N o and the predicted value N p were tested using the t-test for independent samples (Sokal and Rohlf 2012): where N i is the mean, s 2 i is the variance, and n i is the size of the ith sample.The statistic t represents Student's t-distribution with n 1 + n 2 -2 degrees of freedom.

Analysis of colonization of dead tree stems
In the wood of all dead trees without bark, distinct holes of T. lineatum were visible.The bark on the thinner part of the stem was difficult to remove completely from the wood without damaging the holes of other bark beetle species, mainly T. minor.T. lineatum had colonized all the dead trees that were analyzed.In 2009 and 2020, colonization variables did not differ significantly: 97.1 holes/m 2 ± 75.3 SD, n = 30 and 91.3 holes/m 2 ± 71.4 SD, n = 8.The main co-occuring species was T. minor (53.5 egg galleries/m 2 ± 44.5 SD, n = 30 in 2009, and 45.6 egg galleries/m 2 ± 38.3 SD, n = 8 in 2020).Egg galleries of T. piniperda were found sporadically in four trees with a mean density of 0.71 eggs gallery/m 2 ± 2.57 SD, n = 4. Holes from T. lineatum were present mainly in the thicker part of the dead tree stems, up to 12 m of their length, and the number of holes decreased with increasing distance from the zone of maximum infestation (Fig. 2).

Relationships between number of T. lineatum holes in meter-long stem units and total number of T. lineatum holes in the entire stem of dead trees
The results of the correlation and regression analyses indicate statistically significant, linear relationships between the number of T. lineatum holes in the distinguished meter-long stem units and the total number of T. lineatum holes in the pine stem (Table 1).The strongest correlations were found for the fourth unit.The coefficient of determination was significant and equals 0.9466, while the mean relative error of estimation did not exceed 19%.Because the distribution of the residual values did not follow a normal distribution (W = 0.9008, P < 0.05; Shapiro-Wilk's test), piecewise linear regression (PLR) was used to achieve a better fit of the regression function.To describe the total number of T. lineatum holes on stems, two PLR models were constructed, comprising two linear equations (model I, Eq. 10) and three (model II, Eq. 11) (Table 2, Fig. 3).
(10) y = −253.619+ 5.562n 4 for n 4 ≤ 420 778.478 + 3.025n 4 for n 4 > 420 The coefficients of determination for the models were significant, with values close to 1 (R 2 = 0.9787 for model I, R 2 = 0.9862 for model II; Table 2).The slopes all differed from zero.For model I, slope a 0 was negative, and slope b 0 was positive.For model II, slopes a 0 and c 0 were positive, while b 0 was negative.The total loss function value for model II, 427,765.3, was about 65% of the value for model I (660,776.1).The breakpoint (α 1 ) for model I was at 420, and the breakpoints for model II were at α 1 = 150 and α 2 = 395 (Table 2, Fig. 3).
The total loss function values Indicate that model II provides a better fit to the data and thus greater precision in assessing the number of T. lineatum holes.In model II, the distribution of residual values with respect to the predicted values is homogeneous over the whole range of values of total number of T. lineatum holes in stems (Fig. 4a), and the residual values followed a normal distribution (W = 0.9777, P = 0.7628; Shapiro-Wilk's test).The relationship between the observed the number of T. lineatum holes in stems and predicted by the model does not deviate significantly from linear (Fig. 4b).These results suggest that the regression ( 11

Accuracy of the method
Using the data from 2020 (data set D3), the accuracy of the method was verified in two ways: (1) for individual dead trees, and (2) for the sample of dead trees (Table 3).Generally, the errors for the observed values (N o ) and predicted values (N p ) for individual dead trees did not exceed 15%, except for trees no. 1 (22.9%) and no. 3 (23.9%).The method was most accurate (relative error less than 5%) for trees with high levels of colonization (dead trees no. 4 and 5).For the sample, the mean observed value N o and predicted value N p did not differ significantly (t test, P > 0.05; Table 3).Based on the analyses of accuracy, it is concluded that the method is highly accurate for estimating the total number of T. lineatum holes.

Selection of study areas
The decomposition rate of dead wood is influenced considerably by the population size and composition of the decomposers.Particularly valuable are data on the size of the decomposer population collected in natural and nearnatural forests.Therefore, the study areas were selected within a protected area (Landscape Park).The high natural value of the Suchedniów-Oblęgorek Landscape Park is reflected in the nature reserves established there (Fig. 1A).
It was necessary to conduct the research in a managed tree stand, because obtaining the data needed to develop the method required a high level of interference in the ecosystem (felling and debarking of a whole tree).The study areas for model validation were more than 2,000 m from the edge of the stand, making it possible to assess the size of the local population of T. lineatum.
The model was validated using the method to assess the amount of lying wood in a forest (e.g., Bobiec et al. 2005).No separate method was developed for standing dead wood (Warren and Olsen 1964;Van Vagner 1968;Caza 1993;Marshall et al. 2000).The key problem in the study was to determine the width of the strip to assess the number of T. lineatum holes.Because lying timber is a section on the horizontal plane, the width of the strip was determined on the basis of the average height of the pine trees in the stand rounded up to the nearest whole tens.The average height of the pine trees is 26 m (Forest Data Bank 2018), and therefore, the width of the strip was 30 m (Fig. 1C).
With the use of a broken line instead of a straight line, a larger area can be assessed.

Evaluation and validation of the model
This study has shown that when the assumptions for linear models are not satisfied, piecewise linear regression (PLR) may be successfully used to estimate the T. lineatum holes.Two models were constructed, consisting of two (model I) or three (model II) linear equations.The fact that the loss function value for model II was about 65% that for model I indicates that the former produces a better fit to the observed data and is thus more useful for estimating the number of T. lineatum holes on dead pines.The greater accuracy of model II is mainly due to the better fit of the regression function for less densely colonized stems (Fig. 3b).The slope of the regression lines in model II indicates a weaker relationship between the number of holes on the fourth stem unit and the total number of T. lineatum holes of the stem.One of the reasons for the beetles' irregular colonization of the thicker part of the stem may be less competition among individuals for suitable resources, which in conditions of low population density are practically unlimited.The colonization of dead trees and biotic interactions was analyzed in detail using data collected in 2009 (D1) by Borkowski and Skrzecz (2016).
The short study period does not allow an accurate assessment of the colonization variables of dead trees by T. lineatum in 2009 and 2020.The similar number of holes in both study seasons could be a random occurrence or indicate ecosystem stability in the park.This remnant of the Świętokrzyska Primeval Forest has been protected for more than 40 years.The eastern part of the park is a large, dense forest complex (Fig. 1A).In Poland, in forests protected in this way, dying and dead trees are left to decompose (Forest Protection Instruction 2012), providing a relatively constant source of breeding material for T. lineatum over the long term.The aforementioned features of the park, combined with the limited forest management operations, may be an important contributor to stabilizing the population of T. lineatum.
The evaluation of the accuracy of the method did not indicate differences between the observed and predicted mean number of holes of T. lineatum on the stems, and the relative errors for individual trees did not exceed 30% (Table 3).These results are evidence of the high effectiveness of the method irrespective of the total number of T. lineatum holes and sample size, which is particularly important in the context of the possibility of using nonlinear estimation (PLR) to evaluate populations of other insect species.Differences in rates of colonization of individual trees have been observed among many groups of forest insects, bark beetles in particular (e.g., Fargo et al. 1978;Bouhot et al. 1988;Borkowski 2019).
Advantages of the developed model are that it does not require a large amount of work or time for data collection and analysis.The total number of T. lineatum holes on sample trees can be estimated by counting the T. lineatum holes on the fourth stem unit (3.0-4.0 m from the stem base).Depending on the purpose for using the method, galleries are counted either after the bark has fallen off naturally (studies of wood decomposition) or after mechanical debarking.The number of holes counted is assigned to the corresponding class (see the Material and Methods section), and the appropriate linear equation (Eq.11, Fig. 3b) is used to calculate the total number of T. lineatum holes on individual pine stems.In addition, the position of the unit on the fourth meter length along the stem counted from the thicker end means that the holes can be counted without the need to fell the tree.Because research is currently carried out by field teams, in the author's opinion the need to count galleries using a 2-3 m ladder is not a significant obstacle to the use of the method.The small amount of time required to apply the model results from the fact that it is constructed based on linear relationships, which means that there is no need for transformations in the data analysis stage, for example.From a practical standpoint, the simple construction of the model enables its use in the field, because it does not require sophisticated statistical procedures.
In recent years in Poland, there have been isolated reports of Trypodendron laeve Eggers (Lukášová et al. 2012), which damages wood similarly to T. lineatum (Bußler and Schmidt 2008).Research in the Świętokrzyskie Mountains has shown that T. laeve, if it occurs in the area at all, is very rare.In 2010, in Świętokrzyski National Park, adjacent to Forest District Zagnańsk, 38 Trypodor pheromone traps distributed evenly across the studied area caught approximately 40,000 bark beetles, and only two were identified as T. laeve (Witkowski et al. 2015).Other studies indicate that it is widespread in Europe, yet apparently rare (Martikainen et al. 1996(Martikainen et al. , 1999;;Martikainen 2000;Kirkendall and Faccoli 2010;Lukášová et al. 2012).Only a few authors consider T. laeve to be common (Krehan and Holzschuh 1999).On the basis of the aforementioned results, it may be assumed that the proposed method to estimate the number holes on dead pines is suitable above all for T. lineatum.

Application of the method for studies of wood decomposition
Dead wood is a very important component of forest ecosystems (e.g., Hanski 2004;Müller and Bütler 2010).The decomposition of dead wood is largely governed by climate, with decomposer groups such as microorganisms and insects contributing to variations in the decomposition rates (Zhou et al. 2007;Rock et al. 2008;Eichenberg et al. 2017;Kahl et al. 2017;Pietsch et al. 2019;Seibold et al. 2021).Many authors point to the important role of biotic interactions between decomposers in this process (Zuo et al. 2014;Bradford et al. 2017;Keiser and Bradford 2017).It should be remembered, however, that all of these analyses and discussions will be fruitful only if we have up-to-date information on the dynamics of decomposers in forests.
T. lineatum is a pioneering decomposers of standing dead pine trees.The method to estimate the number of T. lineatum holes is completely non-invasive and does not interfere with the microhabitat of the trunk.The movement corridors are uninterrupted for organisms living under the bark of the thicker part of the dead tree.Longer retention of the bark on a dead tree not only accelerates the process of decomposition, but also has a positive effect on, among other things, the content of elements in the decomposing wood (e.g., Krankina et al. 1999).Due to the variability in the colonization of individual pine trees, the number of T. lineatum holes should be assessed for each tree separately.
The method for estimating the size of the T. lineatum population can be used (taking into account the T. lineatum population density estimated on the basis of holes) for models to describe decomposition processes in forest ecosystems (e.g., Mäkinen et al. 2006;Rock et al. 2008;Tuomi et al. 2011).

Implications for forest management
T. lineatum colonizes living trees only if they are already significantly weakened; therefore it does not have a direct impact on the health of a stand (e.g., Byers 2004).It acts as a natural selection factor, eliminating the weakest individuals from the population.It reduces the quality of the wood through physical damage caused by the formation of galleries, and discoloration caused by fungal staining of the galleries (e.g., Jankowiak et al. 2017;Lehenberger et al. 2018), causing substantial economic loses.Particularly at risk from T. lineatum is wood in bark stored in the forest immediately after felling.The proposed method may be used as complementary to other methods for the precise estimation of T. lineatum in stands and should be particularly valuable for research related to climate change, which is expected to significantly impact the frequency and severity of disturbances to forest ecosystems (IPCC 2018).When dead and dying pines are present and are colonized by T. lineatum, the method should be used to establish its population in the stand.A measure of the T. lineatum population may be the mean total number of holes on dead standing pine stems in the area analysed, which can be calculated using a representative method (Cochran 1977) as follows: (1) Determine the sample size and sampling scheme (i.e., the procedure to select sample trees such as dead, standing pine).(2) Determine the value of the investigated characteristic for the sample; that is, estimate the total number of T. lineatum holes on the sample trees (the stems of the pines selected in the first step).(3) Estimate the investigated characteristic for the population; that is, compute the mean and confidence intervals for the total number of T. lineatum holes on pine stems in the stand.
The results of the present work may be used as a basis for performing the second step.

Conclusions
Because T. lineatum is a pioneer decomposer of dead standing wood of P. sylvestris, a precise, easy, nponinvasive method, based on statistical principles, was needed to estimate its population.Constant monitoring using the method developed here, carried out in parallel with the monitoring of dead wood in the forest, will make it possible to evaluate the ecological role of this species in the process of decomposition of wood.
A piecewise linear regression (PLR) model using two regression lines does not always describe the analyzed data with sufficient accuracy, particularly in the case of stems that are not densely colonized by T. lineatum.In this case, it is necessary to apply a more complex model based on several lines.Validation of the method indicated that PLR was highly effective, irrespective of the sample size and the total number of T. lineatum holes.The method's flexibility means that it can be used both at tree level (in studies of wood decomposition) and at stand level (in forestry practice).
The method is completely non-invasive; felling or debarking of dead tree stems is not needed, so it can be used even in strictly protected areas.Analogous models should be constructed for the precise estimation of the total number of T. lineatum holes on other species of conifers, especially spruce and fir, which as main forest-forming species comprise a dominant component of ecosystem biomass.
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Fig. 1
Fig. 1 Study region.A Suchedniowsko-Oblęgorski Landscape Park, (a) western and (b) eastern part.R: strictly protected nature reserve.B Example of random selection of sample trees.In the area investigated, 42 dead trees of Pinus sylvestris were present.Trees were selected by simple random sampling without replacement (30 dead trees were randomly chosen).Numbers 1 through 42 in the map are the assigned numbers for each dead tree.C Example of systematic selection of sample trees.Symbols (tree crown and tree sampled) are not drawn to scale ◂

Fig. 2
Fig. 2 Distribution of Trypodendron lineatum on Pinus sylvestris dead trees in 1-m long stem units
equation obtained for model II should be used to estimate the total number of T. lineatum holes in dead trees.