![Histogram with a Y-axis break for the beech regeneration count on the plot for size classes B0, B1 and B2. The proportion of plots without oak regeneration, the mean (x¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{x })$$\end{document} and the variance (σ²) are given](https://www.researchgate.net/profile/Maximilian-Axer/publication/350895490/figure/fig3/AS:1013449043755008@1618636489988/Histogram-with-a-Y-axis-break-for-the-beech-regeneration-count-on-the-plot-for-size_Q320.jpg)
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European Journal of Forest Research (2021) 140:947–968
https://doi.org/10.1007/s10342-021-01377-w
ORIGINAL PAPER
Modelling natural regeneration ofEuropean beech inSaxony,
Germany: identifying factors influencing theoccurrence anddensity
ofregeneration
MaximilianAxer1 · SvenMartens2· RobertSchlicht3· SvenWagner1
Received: 29 June 2020 / Revised: 19 March 2021 / Accepted: 8 April 2021 / Published online: 16 April 2021
© The Author(s) 2021
Abstract
The potential utilisation of natural regeneration of European beech (Fagus sylvatica L.) for forest conversion has received little
attention to date. Ecological knowledge is necessary to understand and predict successful natural regeneration of beech. The
objective of this study was to improve understanding of what drives the occurrence of beech regeneration and, once regenera-
tion is present, what drives its density. In the study, we utilised a forest inventory dataset provided by Sachsenforst, the state
forestry service of Saxony, Germany. The dataset was derived from 8725 permanent plots. Zero-altered negative binomial
models (ZANB) with spatial random effects were used to analyse factors influencing occurrence and density simultaneously.
The results provided by the spatial ZANB models revealed that the probability of the occurrence of beech regeneration is
highly dependent on seed availability, i.e. dependent on source trees in close proximity to a plot. The probability of beech
regeneration rises with the increasing diameter of a potential seed tree and decreases with increasing distance to the near-
est potential seed source. The occurrence of regeneration is affected by overstorey composition and competition exerted by
spruce regeneration. Where sites are affected by groundwater or temporary waterlogging, the impact on the occurrence of
regeneration is negative. Although distance to the nearest potential seed source has an influence on occurrence, this variable
exerts no influence on density. A high regeneration density arises in conjunction with a high beech basal area in the overstorey.
Beech regeneration density, but not occurrence, is negatively affected by browsing intensity. These variables can be used to
predict the occurrence and density of beech regeneration in space to a high level of precision. The established statistical tool
can be used for decision-making when planning forest conversion using natural regeneration.
Keywords European beech· Established natural regeneration· INLA· Zero-altered negative binomial model· Spatial
random effects· Bayesian inference
Introduction
European beech (Fagus sylvatica L.) is the most abundant
broadleaf tree species in central Europe. Due to its high
shade tolerance and very broad ecological amplitude, beech
dominates the potential natural vegetation on many sites and
plays a key role as a late successional species (Ellenberg and
Leuschner 2010).
Natural regeneration can play an important role in the
process of converting pure coniferous stands to mixed stands
by increasing the proportion of broadleaf trees (Felton
etal. 2016), contributing to the restoration of forests. The
Communicated by Derek Sattler.
* Maximilian Axer
maximilian.axer@tu-dresden.de
Sven Martens
sven.martens@smul.sachsen.de
Robert Schlicht
robert.schlicht@tu-dresden.de
Sven Wagner
sven.wagner@tu-dresden.de
1 Chair ofSilviculture, Institute ofSilviculture andForest
Protection, TU Dresden, 01737Tharandt, Germany
2 Department ofForest Management, Forest Inventory
andForest Valuation, State Forest Enterprise Sachsenforst,
01796Pirna, Germany
3 Chair ofForest Biometrics andForest Systems Analysis,
Institute ofForest Growth andForest Computer Sciences, TU
Dresden, 01737Tharandt, Germany
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948 European Journal of Forest Research (2021) 140:947–968
1 3
regeneration phase is the best opportunity to influence tree
species composition and forest ecosystem structures (Fis-
cher etal. 2016; Löf etal. 2018). With natural regeneration
playing an expanding role in silviculture in central Europe,
the monitoring of the success of regeneration in stands is an
important issue for forest management. However, predicting
natural regeneration in mixed stands is difficult (Löf etal.
2018).
To better understand and predict successful natural regen-
eration of beech, a sound knowledge of the key ecological
processes is required. Successful regeneration encompasses
numerous ecological processes such as flowering, seed pro-
duction, seed dispersal, storage, germination, seedling devel-
opment and the successful establishment of juvenile trees
(Harper 2010; Wagner etal. 2010; Fischer etal. 2016).
The process of flowering and the total number of seeds
depends on the trees’ physiological age and the onset of
maturity (Owens and Blake 1985). Within stands, beech
begins seed production between the ages of 50 and 80
(Brown 1953). Wagner (1999) established a relationship
between seed production and tree diameter. Beech masts
irregularly, at intervals of three to ten years (Övergaard etal.
2007). The distance to the parent tree is relevant to seed
dispersal by means of barochory and to the resulting seed
density (Sagnard etal. 2007). While barochory commonly
results in beech nut dispersal up to a radius of 20m (Wagner
1999; Kutter and Gratzer 2006; Millerón etal. 2013), zoo-
chorus dispersal is dependent on the suitability of the stands
as a habitat for seed-dispersing animals (Löf etal. 2018).
Kunstler etal. (2007) found beech regeneration in pine
stands up to 3000m from the nearest mature beech stand.
In temperate latitudes, plants usually bridge the unfavour-
able growth conditions of the winter months by means of
dormancy (Runkle 1989). This is usually associated with the
storage of diaspores on the forest floor. There, the diaspores
are susceptible to harmful fungi, predation and unfavourable
environmental conditions (Jensen 1985; Madsen 1995).
After successful germination, establishment and early
growth, beech seedling development is dependent upon
light, soil moisture, nutrient supply and browsing intensity
(Minotta and Pinzauti 1996). Madsen and Larsen (1997)
identified light availability to be the most important factor
influencing sapling density. Beech seedlings can establish
under a wide range of light conditions (Emborg 1998).
However, the survival rate and growth of beech seedlings
increase with increasing irradiation (Madsen 1994; Minotta
and Pinzauti 1996). The overstorey (Paluch etal. 2019), the
shrub layer and the herbaceous vegetation all reduce the
resources available to beech seedlings. Ground vegetation
affects seedling growth especially severely when grasses
dominate the field layer (Coll etal. 2003, 2004; Lin etal.
2014). In addition to interactions prompted by ground veg-
etation and the overstorey, there are effects of competition
between the regeneration of various tree species. Dobro-
volny (2016) demonstrated an increase in the Norway spruce
(Picea abies L.) density with increasing irradiation and a
superior height growth in comparison with beech from a
light intensity of 20% or more. Furthermore, seedling den-
sity can be reduced by browsing. Beech seedlings can be
browsed by birds, rodents, deer and other herbivores (Mad-
sen 1995; Olesen and Madsen 2008).
The aim of this study was to develop a statistical model
to predict natural regeneration of beech. A statistical anal-
ysis will show important variables influencing natural
regeneration. Using the variables, the occurrence of natu-
ral regeneration can be predicted. This is of great interest
for regeneration ecology as well as for practical forestry, as
nature-oriented forest management favours natural regenera-
tion in terms of cost reduction and site adapted regenera-
tion, especially as a means to facilitate adaptation to climate
change (Zerbe 2002; Dobrowolska 2006; Finkeldey and Hat-
temer 2010; Milad etal. 2013; Vrška etal. 2016; Kalisze-
wski 2017; Polley etal. 2018). There are several approaches
for regeneration modelling, from which two groups can be
distinguished: statistical models and mechanistic (process)
models. While mechanistic models predict regeneration as a
system of biological processes (Nathan and Muller-Landau
2000), statistical models allow for an accurate reproduction
of observed patterns. Within statistical modelling, regenera-
tion can be modelled on different spatial scales. While the
first phases of the regeneration process are often investigated
by modelling the dispersal curve through maximum likeli-
hood methods (Ribbens etal. 1994; Clark etal. 1999) and
point process statistics (Wild etal. 2014) on a small scale,
regeneration establishment models use wide-scale routine
inventory data. Regeneration establishment models are
models that deal exclusively with the establishment phase
of regeneration. The probability of the occurrence and the
density of natural regeneration of beech can be analysed and
predicted using regeneration establishment models.
Studies on regeneration establishment models for beech
are rare (Klopčič etal. 2015; Kolo etal. 2017). Many stud-
ies have tended to focus on the ingrowth of beech, including
recruitment over a certain threshold value, e.g. dbh > 10 or
12cm (Klopcic etal. 2012; Manso etal. 2019; Zell etal.
2019). For the purposes of this paper, established regenera-
tion was defined as beech trees with a height greater than
20cm and a diameter at breast height (dbh) smaller than
7cm. In addition to different spatial scales, established
regeneration can also be analysed on different time scales.
Established regeneration can be quantified either as a regen-
eration density over a measurement threshold at a defined
moment in time or as a regeneration rate, i.e. the number of
trees reaching a threshold value over a certain period of time
(Lexerod and Eid 2005). The former approach is preferable
for silvicultural planning, because a certain regeneration
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949European Journal of Forest Research (2021) 140:947–968
1 3
density is important for management seeking to achieve a
high quality stand in the future.
A special feature of forest inventory data based on grid
samples with permanent plots is the small experimental
area of the sample plot. Most investigations include only
plot information when modelling regeneration density (Li
etal. 2011; Zhang etal. 2012; Shen and Nelson 2018). As
a consequence, spatial patterns of seed availability result-
ing from the plot environment are considered in few stud-
ies (Shive etal. 2018; Monteiro-Henriques and Fernandes
2018). In addition to the crucial issue of small plot size,
inventory datasets contain spatial autocorrelation due to abi-
otic or biotic processes and spatial structures in unobserved
explanatory variables (Bachl etal. 2019). Few regeneration
establishment models have considered spatial correlation
(Dobrowski etal. 2015).
To the best of our knowledge, there have been no
approaches to the modelling of beech regeneration tak-
ing spatial dependence into account. We applied a new
approach to a statistical establishment model using the
example of natural beech regeneration in the lowland
forests of Saxony, Germany. The objective of this study
was to improve understanding of what drives the absence
or occurrence of beech regeneration and, once regenera-
tion is present, what drives its density, by testing a wide
collection of explanatory variables. Six hypotheses were
derived from the knowledge presented above: (1) based
on beech nut dispersal, it was hypothesised that the prob-
ability of the occurrence of regeneration and the density
increase with an increase in the plot-specific basal area of
beech and that there is a decrease with increasing distance
between the plot and the nearest mature beech stand. (2)
It was expected that the probability of beech regeneration
increases with beech diameter, as seed production is
related to tree diameter. (3) Due to the reduced availabil-
ity of resources such as water and light as a function of the
overstorey, it was assumed that beech regeneration density
is affected by the total basal area of the overstorey of the
plot. (4) In addition to the properties of the overstorey, it
was hypothesised that due to interspecific competition in
the understorey, beech regeneration density will decrease
with an increasing density of spruce regeneration. (5)
Beech regeneration density is negatively affected by her-
bivory, as a result of increased mortality. (6) Furthermore,
it was assumed that the occurrence and density of beech
regeneration are affected by the soil characteristics of the
site. An excess of water is expected to affect the occur-
rence of beech regeneration negatively.
Materials andmethods
Study area
The study was conducted in state forest located in northern
Saxony, Germany. The region is characterised by Pleis-
tocene lowlands and their transitions to loess uplands.
The region encompasses eight ‘growth districts’ (forstli-
che Wuchsgebiete, Gauer and Aldinger 2005) (Table1).
Figure1 visualises the geographic range of this study
area. Scots pine is the predominant tree species in both
mixed forests and pure stands (Table1). Climate records
(1981–2010) indicate mean annual precipitation of
between 534 and 796mm, and the elevation ranges from
0 to 600m above sea level (Gauer and Kroiher 2012).
Table 1 Distribution of permanent sampling points (PSP) in the growth areas and characterisation of the growth areas in terms of tree species
composition, altitude, average annual temperature (Temp.) and mean annual precipitation (Precip.) (Gauer and Kroiher 2012)
Growth district Number of PSP Beech
share
(%)
Oak share (%) Spruce
share
(%)
Pine share (%) Altitude (m) Temp. (°C) Precip (mm)
Sächsisch-Thüringisches Löß-
Hügelland 60 3 26 16 14 150–450 8.6 631
Sachsen-Anhaltinische Löß-
Ebene 4 1 13 1 4 75–300 9.1 504
Mittleres nordostdeutsches
Altmoränenland 276 1 9 1 74 75–300 8.9 534
Düben-Niederlausitzer Alt-
moränenland 6487 1 4 2 83 0–300 8.9 597
Lausitzer Löß-Hügelland 174 2 8 28 39 0–600 8.2 700
Oberlausitzer Bergland 7 4 3 75 6 300–600 7.7 796
Westlausitzer Platte und
Elbtalzone 1716 5 12 19 45 150–450 8.8 652
Leipziger Sandlöß-Ebene 1 1 13 1 15 150–300 9.1 569
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950 European Journal of Forest Research (2021) 140:947–968
1 3
Forest inventory andanalysis ofdata
The data used in the study were generated in the context of
a forest inventory and were provided by the Saxony state
forestry service (Sachsenforst). The forest inventory sur-
veyed 8725 permanent sample plots on a grid square with a
200m side length (see also Fig.1). Each plot consisted of
three concentric circles with a 6 and 12m radius. In the 6m
circle, all trees with a dbh ≥ 7cm and < 30cm were meas-
ured, while in the largest circle only trees with a dbh ≥ 30cm
were recorded. In each plot, the location of all trees above
the measurement threshold relative to the plot centre was
recorded, based on distance and azimuth. In addition, tree
species, dbh, age and several height parameters were meas-
ured and recorded.
For each plot, a subplot with a 2m radius (≈ 12.57
m2), located 5m north of the plot centre, was established.
Within each subplot, regeneration of height ≥ 20cm
and a dbh < 7cm was measured. The inventory design
distinguished between tree height/dbh classes for
the regeneration. The first size class B0 included
regeneration of height ≥ 20cm and < 50cm. The second
size class B1 included regeneration of height between 50cm
and < 130cm, whereas the third size class B2 consisted of
trees of height ≥ 130cm. The number of individuals was
recorded separately according to tree species, size class,
browsing damage and whether protected from browsing
damage. A distinction was also made between natural and
artificial regeneration. The maximum number of seedlings
recorded was 15; that is, a theoretical distribution of num-
bers was truncated at 15.
We used the number of naturally regenerated beech seed-
lings in the plot as the dependent variable for every size
class, because it was to be expected that regeneration would
be influenced differently by variables within regeneration
development classes (Žemaitis etal. 2019). The occur-
rence and density of beech regeneration were assumed to be
affected by various variables (see Table2 and hypotheses).
All computations were done using R version 3.4.1 (R
Core Team 2018). The following explanatory variables char-
acterised the overstorey of the plot: quadratic mean diameter
of beech in the overstorey (1.3m, Dq), stand basal area (BA)
Fig. 1 Location of the study
region in northern Saxony,
Germany, and the permanent
sampling plot inventory. Dots
represent the coordinates of the
sample plots. Representation
of the triangulation with 7752
vertices used for the SPDE basis
function representation of the
Matérn Gaussian field. Section
representing the southern part
of the study area
Table 2 Summary statistics of the median values and ranges in parentheses for the explanatory variables in the model fit for plots without beech
regeneration, beech regeneration size class B0, beech regeneration size class B1 and beech regeneration size class B2
Variable Acronym Units Without beech Beech B0 Beech B1 Beech B2
Quadratic mean diameter beech Dqcm 0 (0–113.2) 18.72 (0–140.9) 10.5 (0–140.9) 10.1 (0–114)
Upscaled basal area of a plot BA m2/ha 28.61 (0–85.5) 28.61 (0–69.4) 29.4 (0–66.5) 29.6 (0–66)
Basal area beech BA_B m2/ha 0 (0–52.8) 4.2 (0–60.6) 1.8 (0–50.1) 1.8 (0–50.1)
Basal area spruce BA_S m2/ha 0 (0–81) 0 (0–43.5) 0 (0–38.1) 0 (0–41)
Basal area pine BA_P m2/ha 20.1 (0–74.8) 9.2 (0–56.1) 12.2 (0–60.1) 13.2 (0–55)
Distance mature beech stand (age > 80) D80 m 628.7 (0–24,045) 114 (0–2743) 116 (0–6719) 115 (0–8353)
Buffer beech stand, area 100m stand (age > 80) BUF100 ha 0 (0–22.8) 0 (0–30.9) 0 (0–30.9) 0 (0–30.9)
Buffer beech stand, area 500m stand (age > 80) BUF500 ha 0 (0–40.6) 2.6 (0–40.6) 2.7 (0–41) 2.3 (0–41)
Spruce seedling density B1 Spr_no_B1 n 0 (0–16) 0 (0–15) 0 (0–8) 0 (0–15)
Spruce seedling density B2 Spr_no_B2 n 0 (0–15) 0 (0–4) 0 (0–6) 0 (0–12)
Browsing percent BP % 55 (0–100) 52 (0–100) 47.5 (0–100) 41.8 (0–100)
Humus thickness HT t/ha 104 (4–378) 135 (4–342) 137.5 (4–311) 145 (4–343)
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951European Journal of Forest Research (2021) 140:947–968
1 3
and species-specific basal area are determined on the basis
of each plot. The total basal area for the plot and the spe-
cies-specific basal area were scaled up to hectare level. The
understorey was described by explanatory variables. To test
the influence of established spruce regeneration on beech
regeneration, the density in the plot of spruce of size class
B1 and size class B2 was used (Spr_no_B1, Spr_no_B2).
In addition to the plot-specific explanatory variables,
the following variables from the plot surroundings were
included. In order to estimate seed availability across the
study area, the shortest distance from each sampling point
to the nearest potential seed source (mature beech stand or
stand with admixed beech trees older than 80years) D80
was determined using the gDistance function in R’s rgeos
package (Bivand and Rundel 2019). Another proxy variable
to estimate seed availability was buffer. Within three differ-
ent neighbourhoods (BUF100, BUF250 and BUF500), the
proportion of the area occupied by beech stands (> 80years)
was estimated using the gBuffer function in R’s rgeos pack-
age (Bivand and Rundel 2019). For both variables, beech
stands > 80years were chosen in order to examine the estab-
lished regeneration, which may have established several dec-
ades ago (B2). The age criterion ensures that the surrounding
beech stands had already produced seed at some point in
the past. The location of the beech stands was recorded in
the context of the stand inventory and provided as a spatial
polygons data frame.
Russell etal. (2017) showed that it is essential to consider
the impact of browsing when modelling regeneration. The
influence of browsing was characterised by two variables.
First, browsing percentage (BP) was defined as the ratio
between the number of browsed and the total number of
rowan (Sorbus aucuparia L.) within size classes B0 and B1.
The ratio of browsed rowan seedlings was used as the spe-
cies is widely distributed in the study area and preferentially
browsed by deer (Motta 2003; Götmark etal. 2005). The size
classes were chosen based on the size of the game species.
Rowan regeneration of size classes B0 and B1 occurred in
2100 of the 8725 examined plots. Inventory plots without
rowan regeneration were set as not available (NA). Based
on the 2100 plots, the browsing impact was interpolated for
all plots using inverse distance weighting in R’s gstat pack-
age (Gräler etal. 2016). Second, a factor variable charac-
terising the browsing intensity in a plot was defined. The
study distinguished between non-browsed, simply browsed
and multiply browsed rowan, depending on the frequency of
browsing on a single plant. The highest browsing intensity
reported was assigned to the plot as a factor. If there was
no rowan regeneration in the plot, this was assigned to the
category absence of rowan. A binary variable was used to
indicate whether a plot was fenced in to prevent damage
caused by game.
Elevation, aspect, the nutrient content of the soil and its
moisture level were provided by Sachsenforst as site con-
dition variables. The factor variable was divided into four
factor levels according to the moisture level. A distinction
was made between well-drained, temporarily waterlogged
and groundwater-affected sites. Where no level of water
storage capacity was provided for the site, the fourth factor
level ‘not available’ was used. In order to analyse the influ-
ence of humus layer on beech regeneration, the nitrogen and
carbon content and the total thickness of the humus layer
were provided. These data were derived from a regionalised
soil inventory. All variables describing climate conditions
were retrieved from GIS databases provided by Gauer and
Kroiher (2012).
Data pre‑treatments
In order to avoid errors, the data were tested for their suit-
ability in terms of model building (Zuur etal. 2018). Data
exploration was carried out following the protocol described
in Zuur etal. (2010). Outliers, which were checked with
Cleveland dotplots, were not found. In order to select the
correct distribution function for modelling, the data were
examined for zero inflation. Frequency plots were built for
this purpose, to show the number of observations with a cer-
tain count of beech seedlings. In addition, a variance infla-
tion factor (vif) was calculated for every covariate in order to
detect multicollinearity. Collinearity is defined as the exist-
ence of correlation between covariates (Zuur etal. 2010).
Here, the function corvif was used (Zuur etal. 2009). The
threshold value was set at 3 and covariates with higher vifs
were dropped. Subsequent to variable selection, we searched
for relationships between the explanatory variables and the
response variable by deploying multipanel scatterplots.
Finally, the data were examined for dependency structures.
Due to a small proportion of seed-bearing beech in Saxony
(see also Table1) and its limited seed dispersal and patchy
establishment, local density is likely to be autocorrelated
(Flores etal. 2009). This kind of clustering of regeneration
has the potential to lead to zero inflation, especially when a
regular inventory grid is used. Spatial correlation can also
result from unobserved explanatory variables (Bachl etal.
2019).
Estimating seedling density
Due to the nature of the regeneration process, regeneration
data often follow a zero-inflated and overdispersed count
distribution (Zhang etal. 2012; Shen and Nelson 2018).
The observed count of beech seedlings in the Saxonian for-
est inventory also revealed a very high proportion of zeros
(Fig.2). Different methodological approaches have been
used for regeneration and recruitment modelling, in order
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952 European Journal of Forest Research (2021) 140:947–968
1 3
to account for this special feature of the data. In previous
studies, density was often modelled in a two-step approach.
First, the probability of occurrence is modelled based on
a set of covariates. Second, another equation estimates the
density of regeneration based on the same or a different set
of covariates (Vanclay 1992; Sterba etal. 1997; Qin 1998;
Klopcic etal. 2012; Yang and Huang 2015). Recent papers
used a one-step modelling approach by combining two sepa-
rate estimation processes into a joint distribution of proba-
bilities (Shive etal. 2018; Shen and Nelson 2018; Monteiro-
Henriques and Fernandes 2018; Zell etal. 2019). To take
into account zero inflation and overdispersion, they made
use of zero-inflated distribution models. Zero-inflated Pois-
son (ZIP), zero-altered Poisson (ZAP), zero-inflated negative
binomial (ZINB) and zero-altered negative binomial distri-
butions (ZANB) are distributions commonly used to include
excess zeros in the modelling process (Fortin and DeBlois
2007; Li etal. 2011; Zhang etal. 2012). The major advan-
tage of zero-altered models is that both processes (occur-
rence and density) can be investigated, facilitating ecological
understanding (Cunningham and Lindenmayer 2005).
Following the example of Zuur etal. (2018), the model
was fitted in a Bayesian framework using integrated nested
Laplace approximation (INLA). In general, Bayesian infer-
ence for complex spatial models can be implemented with
simulation methods such as the Markov chain Monte Carlo
method (MCMC). Compared to MCMC, INLA is advan-
tageous in terms of faster computation due to determinis-
tic and accurate approximation by integrals in contrast to
stochastic simulations (Krainski etal. 2019). The package
R-INLA (Lindgren and Rue 2015) in R (R Core Team
2018) provides the INLA method to solve complex models
with spatial random effects. For zero-inflated models, there
is currently no approach within the package R-INLA to
model occurrence depending on the covariates (Zuur etal.
2018). Zero-altered negative binomial generalised linear
models (ZANB) were used in this study for the first time to
model both probability of occurrence {0,1} and density of
beech seedlings (BDi) (
i=1, …,n
) at
n
observations in the
plots, as a function of covariates. The ZANB model simul-
taneously estimates the parameters of a zero-truncated
negative binomial distribution conditional on a Bernoulli
trial. The former applies for the count data, whereas the
latter deals with absence occurrence. The variable
1
{BD
i
>
0}
,
which indicates the occurrence (value
1
) or absence (value
0
) of seedlings, is
1
with probability
𝜋i
,
and conditionally, provided seedlings are present,
BDi
has
the truncated negative binomial distribution given by
Pr(
BD
i
=0
)
=1−π
i
,0≤𝜋≤
1
The parameters
πi
and
𝜇i
in both parts are expanded as
a function of different covariates. Except for the explicitly
hierarchical dependencies, all variables are assumed to be
independent.
First, a base or ‘initial model’ taking no account of spa-
tial dependence was used. This model used only covari-
ates as predictors of occurrence and density. To exam-
ine whether patterns of spatial correlation could be used
as surrogates for unobserved explanatory variables, we
added a spatial random structure to the base model, i.e. a
‘complex model’. We then compared the complex model to
the initial model to determine whether the spatial compo-
nent could contribute to explaining the variation in beech
regeneration occurrence and density. In addition, a model
consisting of only a spatial component was tested, i.e.
‘spatial random model’. This procedure was also applied
to a unimodal Poisson, negative binomial and zero-altered
Poisson model. Preliminary investigations showed that the
complex ZANB model exhibited the best fit to the data
(shown in “Optimal random structure” section). The fol-
lowing formula describes the full ZANB model used to
test the variables:
The term πi is the estimated probability of beech regenera-
tion in the plot, i.e. occurrence, and is modelled via the Ber-
noulli part of the model. The logit transformation is defined
as
logit(
𝜋i
)
=log(
𝜋
i
1
−𝜋
i
)
. The γs correspond to the n coeffi-
cients of the binary model, which are multiplied with each
of the n selected explanatory variables zi. A zero-truncated
negative binomial part of the model is used for the
Pr
(BDi=x∣BDi>0)∝ k+x−1
x
⋯
k
1
(
𝜇i
k+𝜇
i)x
for x=1, 2,
…
BDi
∼ZANB
(
𝜇
i
,𝜋
i
,k
)
E
BDi
=
𝜋i
1−P
0
×𝜇iwhere P0=
k
𝜇
i
+k
k
var
BDi
=
𝜋i
1−P0
×
𝜇2
i+𝜇i+
𝜇2
i
k
−
𝜋i
1−P0
×𝜇i
2
log(
𝜇
i)
=𝛽
0
+x
1i
𝛽
1
+...x
mi
𝛽
m
+u
i
u
=
(
ui
)i=1,…,n
∼N(0,
∑u)
logit(
𝜋
i)
=𝛾
0
+z
1i
𝛾
1
+... +z
ni
𝛾
n
+v
i
v
=(vi)i=1,…,n∼N(0,
∑v)
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953European Journal of Forest Research (2021) 140:947–968
1 3
estimation of nonzero densities µi, where µi is log trans-
formed. The βs correspond to the m coefficients of the count
model, which are multiplied with each of the m selected
explanatory variables xi. For the parameter estimation of the
fixed effects, the default prior distributions of R-INLA, i.e.
normal distributions, were used.
𝛽0,…,𝛽m
and
𝛾0,…,𝛾n∼N(0, 1000)
In addition to the fixed effects, the zero-truncated nega-
tive binomial model estimated a hyperparameter k called
size, which takes overdispersion into account. The default
prior distributions of R-INLA were used for the parameter
estimation of k, i.e. log gamma prior distribution (TableA1).
Weakly informative prior distributions were selected for the
coefficients as a priori knowledge was not available.
The terms ui and vi are random intercepts assumed to be
spatially correlated with mean 0 and covariance matrix ∑ in
the counts and binary parts of the model. It was assumed that
u and v exhibit Markovian behaviour and follow a Gaussian
Markov random field (GMRF). To quantify the associated
covariance matrices of u and v, Matérn covariance func-
tions were used and numerically approximated using SPDE
(continuous domain stochastic partial differential equation).
The SPDE approach provided the Matérn covariance param-
eter, i.e. hyperparameters θ1 = log(τ) and θ2 = log(κ). The
hyperparameter κ quantified the range of the effect as the
distance at which the spatial dependency diminishes. The
hyperparameter τ is a measure of the strength of the spatial
random effects (Lindgren and Rue 2015). R-INLA uses a
normal distribution for the priors of θ1 and θ2 (TableA1).
A dense grid of non-overlapping triangles over the sam-
pling area was needed to solve the SPDE. Therefore, the
finite element approach was applied. The boundary of the
study area and the locations of the plots were used to build
the mesh of non-overlapping triangles using the function
inla.mesh.2d(). max.edge and cutoff were used as arguments
within this function in order to control the shape of the trian-
gles. The maximum edge argument specified the triangles’
maximum allowed edge length for the inner and outer part
of the mesh and was set to 700 (Zuur etal. 2017). The cut-
off argument ensured that there was a minimum allowed
distance between two points. This argument was set to 250.
As the coordinate projection was set up in a UTM projec-
tion, these values were in metres. In total, there were 7752
vertices covering the study area. Figure1 shows part of the
mesh that was used to approximate the spatial field. Spatial
prediction at a specific location within in the GMRF was
straightforward because SPDE provided the approximation
of the entire spatial process. It was just a matter of including
the locations where predictions were required as observa-
tions of missing values in the model (Krainski etal. 2019).
The works of Lindgren and Rue (2015), Zuur etal. (2017)
and Krainski etal. (2019) are recommended for further
information on INLA and the SPDE approaches.
As the influence of the explaining variables was expected
to vary between the different size classes of beech regenera-
tion (see Table2), the models were individually fitted for
each size class. All 16 chosen variables were tested as pre-
dictors in both the count and in the binary part of the model.
The model output could be differentiated in fixed effects
and hyperparameters. The posterior mean and the quantiles
were presented to summarise the posterior marginal distribu-
tion. Analogous to frequentist statistics, a parameter is con-
sidered not significant if the 95% credible interval includes
0 when taking the 2.5% and 97.5% quantiles of the posterior
distribution. Non-significant variables were removed from
the fixed part of the model. The posterior mean indicates the
influence of the parameter.
The Watanabe–Akaike information criterion (WAIC) was
applied for model selection. WAIC consists of two terms
representing the quality of fit and the model complexity.
For further details see Gelman etal. (2014). The model with
the smallest WAIC was chosen. The leave-one-out cross-
validation served as a further factor for model selection, used
to quantify the prediction accuracy of the model. This factor
could also be used to validate the model. R-INLA computed
the so-called conditional predictive ordinates (CPO). It was
defined as the posterior probability of an omitted observa-
tion given all the other data for the model fit. Large values
indicate a good fit of the model to the observed data. The
CPO values are presented in dotplots, and the mean loga-
rithm CPO (LCPO) was calculated as a measure summaris-
ing the values. Lower values of the LCPO indicate a better
fit of the model. The model accuracy of the occurrence part
of the model was evaluated using receiver operating char-
acteristic (ROC) curves. The ROC curve is generated by
plotting the rate of true positive results against the rate of
false positive results at different threshold settings. Values
at 0.5 indicate a fit that is no better than random. Values of
1 indicate a perfect fit.
Results
Characteristics ofbeech regeneration
As expected, a high number of the inventory plots did not
reveal any occurrence of beech natural regeneration (Fig.2).
Approximately 91.5% of all plots contained no beech regen-
eration in the B0 size class. On average, 3.9 individuals of
size class B0 were found in the 744 colonised plots (0.31
ind. m−2). Approximately 92.3% and 93.1% of all plots con-
tained no beech regeneration of size class B1 or B2, respec-
tively. Beech regeneration of size class B1 was found in 676
plots, and the mean regeneration count per colonised plot
was around 2.9 (0.23 ind. m−2). Only 601 plots possessed
beech regeneration of size class B2. The mean beech count
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954 European Journal of Forest Research (2021) 140:947–968
1 3
in plots with beech regeneration of size class B2 was around
3.0 (0.24 ind. m−2). Analyses revealed that natural beech
regeneration included an excessive number of non-occur-
rence (zeros) and demonstrated a large variation around the
mean. A higher count of beech regeneration is observed less
frequently than a low count (Fig.2). In the dataset, a regen-
eration count of 15 constituted an exception. The smallest
size class (B0) exhibited a higher density in the colonised
plots than B1 and B2 (Fig.2).
Optimal random structure
Spatial correlation was expected to occur as the data were
collected on a systematic grid. A comparative analysis
revealed that those models with covariates only, and mod-
els with a spatial term only, exhibited higher WAIC values
than the full model, suggesting a poorer fit for the former.
This result indicated that the spatial relationship between the
plots contained valuable information that is not captured by
environmental covariates alone. As a result, the final ZANB
model contained spatial correlation in the count part as well
as in the binary part of the model. For the sake of a simpli-
fied presentation, the two classes of effects (fixed effects
and spatial random effects) for the occurrence and density
of regeneration are described separately in the following.
However, it should be mentioned that the ZANB model fits
both components simultaneously. All effects presented sepa-
rately are conditional on the rest of the model.
Occurrence model ofbeech regeneration
Many of the covariate data elements originate from the same
systematic grid as the regeneration data and other covariates
were interpolated from wider grids. Therefore, we expected
some collinearity between the different covariates. Accord-
ing to the vif, we found multicollinearity among the buffer
queries of different neighbourhoods and between distances
to nearest potential seed sources at different ages selected.
Only uncorrelated covariates were used for the following
analysis.
Among the three different size classes, different vari-
ables had an impact on regeneration occurrence (Table3).
Plotting probabilities of the occurrence of beech regenera-
tion B0 against different Dq at different humus thicknesses
revealed a positive influence of Dq and a slightly positive
influence of humus layer (Fig.3). Based on an estimated
probability greater than the threshold value of 0.5, Fig.3
indicates that beech regeneration was likely to occur from
a Dq of 41cm and higher. The probability of occurrence
increased with increasing Dq. Distance to the nearest beech
stand had a significant negative impact on the occurrence
of beech regeneration. The probability of the occurrence of
beech regeneration decreased with increasing distance to the
nearest mature beech stand (> 80years). In addition, beech
regeneration was found to be more likely as the propor-
tion of the area occupied by beech within the 500m radius
increased. Figure3 highlights the effect of the distance to
the nearest mature beech stand on different sites. The prob-
ability of beech regeneration decreased drastically when
the distance to the nearest beech stand became greater than
1000m. Beech regeneration was very unlikely at yet greater
distances, as indicated by an estimated probability of 0.34.
Where distance was equal, the probability of the occurrence
of beech regeneration on well-drained sites was greater than
on groundwater-affected and temporarily waterlogged sites.
Figure3 also highlights the importance for the occurrence
of beech regeneration the influence of spruce regeneration
density and spruce basal area. Where there are five spruce
seedlings of size class B2 or more on a plot, beech regen-
eration of size class B0 can no longer be expected to occur.
This corresponds to a spruce regeneration density of 4000
per hectare. The probability of beech regeneration occur-
rence became very low where spruce basal area exceeded
30 m2/ha.
Dq, distance to the nearest mature beech stand, basal
area of spruce, humus thickness and excess water showed
Fig. 2 Histogram with a Y-axis break for the beech regeneration count on the plot for size classes B0, B1 and B2. The proportion of plots with-
out oak regeneration, the mean (
x)
and the variance (σ2) are given
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955European Journal of Forest Research (2021) 140:947–968
1 3
similar effects on the probability of the occurrence of beech
of size class B1 as was the case for size class B0 (Table3).
While the occurrence of beech regeneration of size class
B0 was very likely for a Dq > 41cm, size class B1 occurred
more frequently from Dq > 55cm, indicated by a probabil-
ity ≥ 0.5 (Fig.3). The probability of occurrence decreased
with increasing distance to the nearest mature beech stand.
Regeneration was unlikely on well-drained sites from dis-
tances ≥ 300m. Regeneration was also unlikely on ground-
water-affected sites, even in the vicinity of mature beech
stands, indicated by an estimated probability < 0.5. The
covariates pine basal area and spruce regeneration density
did not show any effect on the occurrence of beech of size
class B1.
In contrast to size classes B0 and B1, the occurrence of
size class B2 was not dependent on spruce and pine basal
area. However, Dq, distance to the nearest mature beech
stand, humus thickness and excess water showed similar
effects on the probability of occurrence (Table3). Beech
regeneration of size class B2 occurred from Dq ≥ 70cm
(see Fig.3). Furthermore, occurrence was only likely in the
vicinity of mature beech stands. The model predicted no
beech regeneration of size class B2 on groundwater-affected
sites, independent of the distance to the nearest mature beech
stands, with an estimated probability of 0.2.
Count model ofbeech regeneration
Stand variables significantly influenced the extent of beech
regeneration (Table4), whereas no site variables exhibited
an effect on regeneration density. Among the stand variables,
beech basal area and Dq had a positive effect on the density
of size class B0. Figure4 shows the effect of beech basal
area with a variable Dq on regeneration density. The quad-
ratic term of beech basal area was found to have a negative
effect on regeneration density of size class B0. Regenera-
tion density rose sharply with increasing beech basal area,
especially for plots > 20 m2/ha. After reaching the maximum,
the regeneration density of size class B0 decreases as the
beech basal area increases. In addition, beech regeneration
density is higher with higher Dq. A high density of beech
regeneration of size class B0 was more likely in plots with
non-browsed or simply browsed rowan. Plots with the fac-
tor multiply browsed rowan had a significantly lower beech
regeneration density than those with only low or no brows-
ing of rowan. The density of beech regeneration decreased
with increasing basal areas of pine and spruce. Spruce
basal area had a significant negative influence on density.
The model predicted a low beech regeneration density for a
spruce basal area of 30 m2/ha (Fig.4).
Beech basal area, Dq and pine basal area showed similar
effects on the density of beech of size class B1 as was the
case for size class B0. However, the model did not reveal
any effect of spruce basal area on the regeneration density of
size class B1. Whereas the factor browsing intensity showed
an impact on the regeneration density of size class B0, the
browsing percent had a significant negative effect for size
class B1. Figure4 highlights the effect of Dq and browsing
percent on the regeneration density of size class B1. Beech
regeneration density was found to be higher for plots with
a low browsing percent in rowan. The regeneration density
of size class B1 was lower than that of size class B0. The
maximum predicted regeneration density was 11 seedlings
per plot, corresponding to a density of 8750 seedlings per
hectare (Fig.6).
Beech basal area and Dq had similar effects on regenera-
tion density for size class B2 as for size classes B0 and B1.
Table 3 Fixed effects of the occurrence/absence part (Bernoulli part)
of the ZANB models for size classes B0, B1, B2
The optimised models with the lowest WAIC values are presented. In
the table, the output is shown with the posterior mean and the 95%
credible intervals (0.025 and 0.975). Interpreting the estimated val-
ues, it should be noted that the covariates have quite different ranges
Model Covariate Mean 0.025 quant 0.975 quant
B0 Intercept − 2.323 − 2.854 − 1.809
Dq0.050 0.045 0.055
D80 − 0.001 − 0.001 − 0.001
Spr_no_B2 − 0.331 − 0.737 − 0.013
BA_S − 0.023 − 0.041 − 0.007
BA_P − 0.013 − 0.020 − 0.006
HT 0.003 0.001 0.005
BUF500 0.025 0.007 0.043
Groundwater-affected − 0.647 − 1.113 − 0.171
Well-drained − 0.072 − 0.075 0.347
Temporarily water-
logged − 0.693 − 1.257 − 0.130
B1 Intercept − 2.722 − 3.262 − 2.204
Dq0.033 0.028 0.037
D80 − 0.001 − 0.002 − 0.001
BA_S − 0.021 − 0.038 − 0.004
HT 0.003 0.001 0.005
Groundwater-affected − 0.726 − 1.235 − 0.204
Well-drained 0.237 − 0.183 0.686
Temporarily water-
logged − 0.002 − 0.559 0.563
B2 Intercept − 2.889 − 3.457 − 2.340
Dq0.028 0.023 0.032
D80 − 0.001 − 0.002 − 0.001
HT 0.004 0.002 0.006
Groundwater-affected − 0.938 − 1.466 − 0.400
Well-drained 0.030 − 0.403 0.485
Temporarily water-
logged − 0.742 − 1.374 − 0.115
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956 European Journal of Forest Research (2021) 140:947–968
1 3
In contrast to the effects they had on size classes B0 and B1,
spruce and pine basal area had no significant effect on size
class B2. The total basal area in a plot did influence density
negatively, however (Fig.4). Similar to the effects it had
on size class B1, browsing percent in rowan had a negative
impact on regeneration density. The predicted density for
size class B2 was even lower than for size classes B0 and B1
at a medium beech basal area (Fig.4).
Spatially smooth random effect
In addition to the fixed effects, the model included spatial
random effects. The mesh used to fit the data is presented in
Fig.1. The spatial random field was defined by the hyperpa-
rameters θ1 and θ2. TableA2 (Appendix) shows the values
of the hyperparameters that defined the spatial range and
variance of the random field. In order to visualise the spatial
random effects, an example area within the study area was
selected. It is a representative large closed forest area used
to explain the spatial random effects, which were estimated.
Figure5 shows the example area of the spatial ran-
dom field for the binary part of the model, Fig.6 for the
zero-truncated count part. The spatial random fields for
the Bernoulli part of the models indicate that there was an
obvious spatial pattern of unexplained variability in the
example area where the probability of beech occurrence
could not be precisely predicted by the fixed effects alone.
The probability of the occurrence of beech regeneration of
size class B0 was higher to the south of the example area
in reality than was modelled by the fixed effects. The prob-
ability of occurrence was higher especially at the edges of
fragmented forest areas, both in the example area (Fig.5)
and in other parts of the study area. For some parts of
the north-west of the study area, the probability of regen-
eration occurrence was also higher than predicted by the
fixed effects. Furthermore, there were parts with a high
probability of an absence of regeneration. In the case of
the example area in Fig.5, the probability of occurrence
was higher in the northern part. In the eastern part of the
example area, the probability of occurrence was lower than
predicted by the fixed effects. Spatial random effects for
the occurrence of beech regeneration for size classes B1
and B2 showed similar but more distinct effects than for
size class B0.
Fig. 3 Predicted probability of the occurrence of natural regenera-
tion of beech depending on Dq and humus thickness (left), distance to
the nearest mature beech stand on different sites (middle) and spruce
basal area with varying numbers of spruce seedlings B2 (right) for
different size classes (all continuous variables are set to their mean;
all factors are set to their most frequent factor level). The confidence
intervals were not plotted for reasons of clarity, as they are quite large
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957European Journal of Forest Research (2021) 140:947–968
1 3
The precision of the prediction of occurrence achieved
by the different models is shown both in Fig.5 and in the
ROC curves (Figure A1, Figure A2 and Figure A3). These
figures also show how the model prediction is improved by
the random effects, as a comparison is presented including
and excluding random effects.
The spatial random fields for the zero-truncated count
part of the models is shown(Fig.6), using the same example
area referred to above. The figure reveals a pattern of unex-
plained variability in parts of the example area where beech
regeneration density was not predicted precisely by the fixed
effects alone. The comparison of the random fields indicated
that the random effects were smaller for size class B0 than
for B1 and B2. There were some areas for which density
was higher than predicted by fixed effects. For the B2 size
class model, the uis were even close to 1.6, corresponding
to a fourfold increase in density due to spatial correlation.
Most of these areas were at the edges of the forest com-
plexes and close to villages and the city of Dresden. The
observed regeneration density was high in these locations.
However, there were some areas exhibiting a density 60%
lower than predicted by the fixed effects. Most of these areas
were inside the forest areas (Fig.6). The precision of the
overall prediction provided by the different models can be
seen both in Fig.6 and in the CPO values (Figure A4, Figure
A5 and Figure A6). The comparison of observed densities
with predicted densities, root mean square error (RMSE) and
bias estimate in the form of the mean prediction error (E), as
well as the CPO values of the leave-one-out cross-validation
indicated good prediction accuracy (TableA3).
Discussion
Methodological discussion
In this study, we successfully created complex ZANB spa-
tial models with various covariates stemming from both the
plot and the surrounding stand. In contrast to many earlier
studies that followed a frequentist approach (Zhang etal.
2012; Klopcic etal. 2012; Yang and Huang 2015; Shen and
Nelson 2018; Monteiro-Henriques and Fernandes 2018; Zell
etal. 2019), a Bayesian approach was applied. Although
maximum likelihood methods and Bayesian methods with
weakly informative priors provide the same results for fixed
effect models (Rue etal. 2009; Zuur etal. 2018), the great
advantage of using Bayesian methods in R-INLA is the user-
friendly integration of spatial random effects. The definition
of complex variance structures (e.g. Matérn covariance func-
tion) is straightforward. The MCMC algorithm (Rathbun and
Fei 2006; Flores etal. 2009; Entezari etal. 2019) and the
INLA algorithm can be used to estimate the spatial random
effects of geostatistical models. However, the computational
efficiency of INLA means it exhibits much higher computa-
tion speed than MCMC approaches (Rue etal. 2009; Lind-
gren and Rue 2015). In addition, R-INLA offers a simple
way to provide criteria for model comparison and prediction
accuracy (Rue etal. 2009).
The ZANB model provides a useful framework for mod-
elling regeneration data with an excess of zeros. It enabled
us to examine both processes relevant to occurrence and
processes that influence density, and therefore improve our
understanding of the regeneration ecology of beech (Welsh
etal. 1996; Cunningham and Lindenmayer 2005). The two-
step computation described in chapter2.4. is how Zuur etal.
(2018) suggested applying the function inla(), which cannot
directly handle unequal probabilities
𝜋i
. This was done in
one pass by applying the function to a stacked response vec-
tor consisting of all
1{BDi>0}
,
i=1, ..., n
, and the nonzero val-
ues
BDi
, and approximating the truncated negative binomial
distribution for the second part with a zero-altered negative
binomial distribution where the probability of zero values is
fixed at a number very close to
0
. However, not all probabil-
ity distributions are currently available for model building in
R-INLA. There is no possibility within R-INLA to calculate
Table 4 Fixed effects of the count part (zero-truncated) of the ZANB
models for size classes B0, B1, B2
The optimised models with the lowest WAIC values are presented. In
the table, the output is shown with the posterior mean and the 95%
credible intervals (0.025 and 0.975). Interpreting the estimated val-
ues, it should be noted that the covariates have quite different ranges
Model Covariate Mean 0.025 quant 0.975 quant
B0 Intercept − 0.141 − 0.458 0.156
BA_B 0.073 0.038 0.107
I(BA_B^2) − 0.001 − 0.002 − 0.001
Dq0.018 0.011 0.025
BA_S − 0.049 − 0.080 − 0.018
BA_P − 0.016 − 0.026 − 0.005
BI_without 0.253 − 0.127 − 0.648
BI_simple 0.766 0.116 1.459
BI_multiple − 0.452 − 0.835 − 0.066
Size k0.645 0.617 63.110
B1 Intercept − 0.135 − 0.546 0.256
BA_B 0.023 0.004 0.043
Dq0.019 0.011 0.027
BA_P − 0.02 − 0.032 − 0.008
BP − 0.006 − 0.011 − 0.001
Size k0.447 0.346 0.511
B2 Intercept 0.327 − 0.218 0.857
BA_B 0.044 0.021 0.068
Dq0.014 0.005 0.024
BP − 0.009 − 0.015 − 0.004
BA − 0.023 − 0.036 − 0.010
Size k0.368 0.352 0.399
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958 European Journal of Forest Research (2021) 140:947–968
1 3
a zero-inflated negative binomial model with covariates in
the occurrence part of the model (ZINB) (Zuur etal. 2018).
The response variable was formed by the occurrence
and density of naturally regenerated beech seedlings above
20cm in height and below 7cm in dbh. This choice involved
several restrictions. Smaller seedlings of below 20cm height
were not modelled as they were not recorded in the Saxony
state forest inventory. Therefore, it may be deemed uncer-
tain whether our model is able to represent the first stages
of regeneration establishment. However, a clear advantage
of a height threshold of 20cm is that the most critical phase
in the life cycle of the plant, namely germination and first
establishment, has already been overcome, and thus, the
information about these plants has relevance for manage-
ment. In contrast, during germination, beech seedlings can
be damaged by drought, frost or biotic pathogens (Jensen
1985; Minotta and Pinzauti 1996; Coll etal. 2003), making
information about germinants and very small seedlings less
valuable for management.
Our models for beech regeneration were based on for-
est inventory data. Inventory data are suitable for regenera-
tion modelling as the data is representative for the sampled
area and a large number of plots covers wide gradients of
covariates. In addition, the model can be improved after the
next inventory with follow-up data. A major disadvantage
of inventory data is the small experimental plot size. As a
consequence, the dispersal of beech cannot be depicted by
the plot itself. The distance to the nearest mature beech stand
is an estimate of seed availability. It is a biased estimator of
the effective dispersal distance providing underestimations
of the dispersal distance, as has been shown by parentage
analysis (Oddou-Muratorio etal. 2011; Millerón etal. 2013).
However, dispersal can be overestimated if not all mature
beeches have been recorded. The accuracy of the detection
of beech stands in the stand inventory is crucial. Further
information on this topic can be found in Axer and Wagner
(2020).
Inventory data used in regeneration modelling usually
have a nested structure and are grouped within geographic
locations. In addition, regeneration is a rather uncertain pro-
cess, as it is affected by successful flowering, masting, stor-
age, germination and establishment. The regeneration data
exhibited a high degree of spatial autocorrelation, which
should be considered when modelling natural regeneration.
Few satisfactory results have been obtained for regenera-
tion establishment models and their predictions in the past
Fig. 4 Predicted density of natural regeneration of beech depending
on beech basal area with varying quadratic mean beech dbh (left),
beech basal area with browsing intensity and percent in rowan (mid-
dle), pine basal area with different spruce basal area (middle) and
total basal area for different size classes (all continuous variables
are set to their mean; all factors are set to their most frequent factor
level). The confidence intervals were not plotted for reasons of clar-
ity, as they are quite large
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959European Journal of Forest Research (2021) 140:947–968
1 3
(Schweiger and Sterba 1997; Wikberg 2004; Klopčič etal.
2015). Miina etal. (2006) recommended including random
effects in the modelling approach. It could be shown that
taking random effects (Li etal. 2011; Welch etal. 2016;
Shive etal. 2018; Zell etal. 2019) or spatially correlated
residuals into account (Rathbun and Fei 2006; Flores etal.
2009) increases the model adjustment and improves pre-
diction. Compared to plot-specific or stand-specific random
effects, modelling fixed effects with spatially continuous
random fields provides more interpretable results of the
random effects and provides insights into the spatial correla-
tion of regeneration data. The applied method has also been
employed successfully in ecology research (Beguin etal.
2012; Leach etal. 2016; Dutra Silva etal. 2017; Sadykova
etal. 2017), climate research (Cameletti etal. 2013) and
medical research (Musenge etal. 2013; Arab 2015; Blangi-
ardo etal. 2016). This suggests a theoretical suitability for
regeneration establishment models also, as spatially corre-
lated count data are also present here.
Spatial prediction for a given location is straightforward
as the model provides the approximation of the entire Gauss-
ian Markov random field (Lindgren and Rue 2015). The var-
iability is correctly assigned to the predictions and enables
predictions at unsampled locations (Beguin etal. 2012). The
prediction in INLA can be performed by including locations
in the model with missing response values. In this way, maps
for the occurrence and density of beech regeneration can be
generated easily (Fuglstad and Beguin 2018; Krainski etal.
2019). This is an important planning tool for forest manage-
ment (Figs.5, 6). It supports decision-making on how to
utilise natural beech regeneration, e.g. for forest conversion.
Established regeneration in inventory plots, that is, trees
at least 3–5years old, may have undergone varying ecologi-
cal conditions from the time of germination and establish-
ment until the day of inventory. Due to the time lag between
Fig. 5 Example area from within the study region with observed
occurrence of beech regeneration (left), spatial prediction of the
occurrence of beech regeneration (middle) and spatial random field
for the binary part of the ZANB model (right) for the three size
classes. Areas shaded dark blue have a lower probability of occur-
rence of regeneration than that predicted by the fixed effects. Yellow
to red areas have a higher probability
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960 European Journal of Forest Research (2021) 140:947–968
1 3
early establishment and the date of the inventory, the effect
intensities of the explanatory variables employed may not
be as strong as in studies with a coordinated examination
design, e.g. true experiments. The variables used for our
model are all ecologically interpretable. Other studies have
used, for example, forest ownership as an explanatory vari-
able for regeneration establishment (Kolo etal. 2017), which
has no apparent ecological meaning, or its relevance is
obscured. In the following, the fixed effects will be discussed
in an ecological context and related to the regeneration cycle
of beech (Wagner etal. 2010; Fischer etal. 2016).
Variables influencing theoccurrence ofbeech
regeneration
The occurrence of beech regeneration is influenced by
both stand and site characteristics. Due to the short mean
seed dispersal distances of beech (Wagner 1999; Millerón
etal. 2013), the occurrence of mature beech in a plot is
the primary driver of the occurrence of beech regenera-
tion. As mean dbh offers reliable evidence of fructifica-
tion (Wagner 1999), the finding that the quadratic mean
dbh of beech influences regeneration occurrence fits. Kolo
etal. (2017) arrived at a similar result. They found a posi-
tive influence of the mean tree height and the mean age of
beeches in the overstorey, which are both closely related
to the dbh. A sequence over time can be observed in the
explanatory variable as higher levels of beech regeneration
are predicted at a larger mean beech dbh. For quadratic mean
diameters > 41cm, the probability of beech regeneration of
size class B0 is higher than 0.5 (Fig.3). In the case of beech
regeneration of size class B2, regeneration occurred several
years ago and during this time the mature beech trees could
grow from 40 to 70cm dbh.
However, beech regeneration was also found in plots
lacking mature beech trees. Due to the small plot sample
Fig. 6 Example area from within the study region with observed den-
sity of beech regeneration (left), spatial prediction of beech regenera-
tion density (middle) and spatial random field for the zero-truncated
count part of the ZANB model for the three size classes and part of
the investigation area. Areas shaded dark blue have a lower probabil-
ity of high regeneration density than predicted by the fixed effects.
Yellow to red areas have a higher probability. (Color figure online)
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
961European Journal of Forest Research (2021) 140:947–968
1 3
size, it was not possible to depict seed distribution com-
pletely. Spatial patterns of seed availability resulting from
the plot environment have only been taken into account in a
few studies to date (Shive etal. 2018; Monteiro-Henriques
and Fernandes 2018). The model indicated that distance
to seed source is another major driver (and limitation) of
beech regeneration patterns. This finding was also observed
in other studies (Kunstler etal. 2007; Irmscher 2009; Pol-
janec etal. 2010; Millerón etal. 2013). Poljanec etal. (2010)
found that an increase in the distance to the nearest beech
stand of 1000m reduced beech ingrowth (> 10cm dbh) by
46%. Our results also confirmed that the probability of the
occurrence of regeneration of size class B0 drops to below
0.5 at a distance of 900m (Fig.3). When modelling estab-
lished regeneration data, the effective dispersal distance is
used. The distances estimated using our model are, there-
fore, distances for the effective dispersal. These are not to
be equated with distance estimates that are based on the
primary dispersal, i.e. barochorus seed dispersal (Wagner
1999; Kutter and Gratzer 2006; Sagnard etal. 2007). In the
case of effective seed dispersal, the germination and survival
of regeneration has to be taken into account. Millerón etal.
(2013) showed, for example, that the effective dispersal of
beech exceeds the primary dispersal. According to fructi-
fication of beech, the age selection of stands ≥ 80years for
distance determination showed the best fit. The predicted
probability of the occurrence of beech of size classes B1 and
B2 was smaller than for B0 at the same distance. The differ-
ent probabilities of occurrence for the different size classes
(Fig.3) contradicted the coefficients, which were estimated
to be the same by the model for all size classes (Table3).
The different probabilities in Fig.3 can be explained by the
different total density between the size classes. The thresh-
old of 1 regenerated beech {0;1} was reached due to the
lower plant density for B1 and B2 at shorter distances. The
lower plant density in the larger size classes was a conse-
quence of density-dependent and density-independent mor-
tality with increasing growth (Akashi 1997; Collet and Le
Moguedec 2007).
There are two mechanisms for seed dispersal of beech
given the two variables in question, namely occurrence
of beech in the plot overstorey and distance to the near-
est beech stand in the plot environment. While close-range
seed dispersal is determined by barochory, zoochorus seed
dispersal becomes decisive at distances ≥ 20m. Compared
to the density of seed dispersed barochorusly, the density of
seeds dispersed by zoochory is much lower. Millerón etal.
(2013) emphasised that zoochorus dispersal takes place
on two scales. While small mammals, especially mice, are
only able to transport beech nuts over small to medium dis-
tances of up to approximately 30m (Jensen 1985; Jensen
and Nielsen 1986; Den Ouden etal. 2005), birds can move
seeds over longer distances (Nilsson 1985). Observed distant
dispersal events were probably due to activities of the jay
(Garrulus glandarius L.) (Turcek 1961; Nilsson 1985). The
nuthatch (Sitta europaea L.) has smaller territories than the
jay and usually deposits seed near the seed source and rarely
at distances greater than 40m. Nilsson (1985) observed jays
transporting beech nuts about 1km from the seed source.
Gómez (2003) noted an average distance of 250m from the
seed source for holm oak acorns (Quercus ilex L.). Jays hide
acorns and probably beech nuts in places that are suitable for
germination and establishment (Bossema 1979). Although
only a small proportion of the beech mast is hidden by the
jay, these seeds seem to have a relatively high survival rate
(Nilsson 1985). However, Perea etal. (2011) emphasised
that for the dispersal of beechnuts by jays, the occurrence of
other preferred species, e.g. oaks, and the frequency of non-
masting years should be considered. Nilsson (1985) stated
that jays only hide beech nuts when there are no acorns in a
particular year. In the case of American beech (Fagus gran-
difolia EHRH.), dispersal distances of up to 4km from the
mother tree have been recorded for the blue jay (Cyanocitta
cristata L.) (Johnson and Adkisson 1985). The results in
relation to the influence of the distance to nearest mature
beech stand (D80 in Table3) may reflect the behaviour of
the jay.
A high basal area of spruce in a plot reduces the prob-
ability of the occurrence of regeneration of size classes B0
and B1. Beech regeneration in spruce stands is limited by the
high stand density, a lack of mother trees and a large distance
to neighbouring beech stands. Poljanec etal. (2010) showed
that beech ingrowth was not possible in stands featuring a
spruce share of 70%.
The negative impact of spruce regeneration density on
beech regeneration was an indication of interspecific com-
petition. Existing spruce regeneration of size class B2 pre-
vented establishment of beech seedlings. In contrast, spruce
regeneration of size class B2 revealed no effect on beech
regeneration of size classes B1 and B2. Investigations by
Unkrig (1997), Dobrovolny (2016) and Kühne and Bartsch
(2003) examined the influence of irradiation on the compo-
sition of beech–spruce regeneration. A higher regeneration
density of spruce was associated with higher irradiation due
to canopy gaps. Therefore, a higher amount of irradiation
can be assumed to promote the establishment of spruce
regeneration and, hence, prevent the establishment of beech
regeneration.
The effect of site suggests that regeneration is not to be
expected on sites that are not favourable for beech. This find-
ing is consistent with the findings of Poljanec etal. (2010).
They found a higher probability of beech occurrence on sites
with a higher proportion of beech in the potential natural
vegetation, which considers soil conditions as well. The
absence of beech on hydromorphic sites is an empirical phe-
nomenon that has been observed repeatedly (Ellenberg and
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962 European Journal of Forest Research (2021) 140:947–968
1 3
Leuschner 2010; Aertsen etal. 2012). For the development
of an intensive, deep-reaching root system, beech requires
loose, well-aerated soil (Asche etal. 1995). In pseudogley
soils, it is almost exclusively the fine roots that penetrate
the compacted horizons, along shrinkage cracks (Kreutzer
1961). Schmull and Thomas (2000) were able to demonstrate
morphological reactions of beech seedlings to waterlogging.
Seedlings cannot generate roots below the water table. In
addition to morphological reactions, physiological reactions
that indicate increased stress in plants can be shown. This
finding is consistent with our results. The negative influence
of hydromorphic sites increases with increasing size class
(Fig.3).
Variables influencing beech regeneration density
Taking the ecological characteristics of beech into account,
beech regeneration density is highly dependent on the dis-
persal of beech nuts (Sagnard etal. 2007; Millerón etal.
2013). Most beech nuts fall within the vicinity of the seed
tree. According to earlier studies (Poljanec etal. 2010;
Klopcic etal. 2012; Ramage and Mangana 2017; Žemaitis
etal. 2019), the basal area of beech in a stand significantly
influences the density of beech regeneration. A high den-
sity of beech regeneration is more likely when the beech
basal area in a plot is high. A larger proportion of beech
in the plot leads to more beech nuts and, consequently, to
more beech regeneration. As beech nuts are highly attrac-
tive to seed predators, the rate of predation of beech nuts
by small mammals in stands with a low degree of beech
admixture could be higher (Kelly and Sork 2002; Yasaka
etal. 2003; Kon etal. 2005). The lower beech regeneration
density for admixed beech trees can, therefore, result from
reduced seed availability and effective predation. In addi-
tion to beech basal area, Dq is positively related to beech
regeneration density because seed production is related to
diameter (Wagner 1999). Furthermore, beech is a late suc-
cessional species, able to regenerate and establish under a
wide range of light conditions (Emborg 1998). This was
evidenced by the fact that the size classes B0 and B1 were
not dependent on the total basal area. However, there was
an optimum relationship between beech basal area and the
density of seedlings of size class B0. Different studies have
highlighted the weak relationship between regeneration den-
sity and light availability (Madsen and Larsen 1997; Szwa-
grzyk etal. 2001; Petritan etal. 2007; Ammer etal. 2008).
Ammer etal. (2008) revealed that the importance of light
availability increases with seedling size. All of these results
are consistent with the natural regeneration methods com-
monly used for beech. A shelter provided by mature trees
combined with a long regeneration period leads to a high
regeneration density (Nyland etal. 2006; Peña etal. 2010).
Although the distance to the nearest mature beech stand
had a positive influence on the occurrence of beech regen-
eration, it did not have any effect on beech regeneration
density. Compared to barochorus seed density, the density
of zoochorously dispersed seeds was significantly lower.
Various investigations of long-distance dispersal of beech
seed (Kunstler etal. 2007; Irmscher 2009; Dobrovolny and
Tesař 2010; Mirschel etal. 2011) revealed only very low
regeneration densities at higher dispersal distances. There-
fore, the distance to the nearest mature beech stand influ-
ences only occurrence {0;1}, but not density {1,…15}.
Kunstler etal. (2007) found an average density of 52 beech
seedlings per hectare in pine stands. Dobrovolny and Tesař
(2010) observed a density of 65 plants per hectare at a dis-
tance of 200m from a seed source, while Irmscher (2009)
detected 5 beech seedlings per hectare at 240m from the
nearest seed source. According to our sampling design, a
count of 1 beech seedling per plot corresponded to a den-
sity of 795 per hectare. As can be seen in Axer and Wagner
(2020), higher densities are almost completely absent at
greater distances.
Several studies have shown the impact of browsing on
regeneration (Ammer 1996; Motta 2003; Boulanger etal.
2009). The results of our study demonstrated that browsing
intensity and browsing percentage in rowan had no effect on
the occurrence of beech regeneration in this study area, but
there was an influence on density. The browsing intensity in
rowan was taken as an indication of a certain concentration
of game (Chevrier etal. 2012). In contrast to Kamler etal.
(2010), who found no correlation between deer density and
browsing intensity in rowan, we were able to demonstrate a
spatial variation in browsing intensity in rowan, indicating
a concentration of game. The overall game density would
appear to be decisive for these contrasting findings. Our
results reinforced the findings of Klopcic etal. (2012) and
Russell etal. (2017). Both of these studies found no impact
of browsing on occurrence, but on density. Compared to
other tree species, beech is less severely affected by brows-
ing. This makes an absence of beech regeneration due to
browsing unlikely (Motta 2003; Götmark etal. 2005; Bou-
langer etal. 2009). However, density is indirectly influenced
by browsing, as the mortality rate of regeneration increases
and other plants not preferentially browsed can grow better.
Surprisingly, plots with the factor simply browsed rowan
exhibited a higher density of beech regeneration of size
class B0 than plots with non-browsed rowan. This finding
was consistent with the insights obtained by Kupferschmid
(2018), who found that browsing of a certain species also
depends on the occurrence of other tree species. The occur-
rence of rowan can lead to a decrease in browsing of beech.
In line with our results, Mirschel etal. (2011) also found a
positive correlation between browsing intensity and beech
regeneration.
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963European Journal of Forest Research (2021) 140:947–968
1 3
As has been demonstrated, browsing affects regeneration
density. According to Kolo etal. (2017), browsing is a major
source of error in predicting the probability of regeneration.
Therefore, they recommended that browsing intensity be
considered in future studies.
Random effects
Many ecological models take into account spatial autocor-
relation in data that is due to biotic and abiotic processes
(limited seed dispersal, clumping of safe sites) as well as
spatial structure in unobserved explanatory variables (Flores
etal. 2009; Bachl etal. 2019). However, the variation of the
mean of the random field expresses the variation remain-
ing after taking into account the effects of the covariates
(Dutra Silva etal. 2017), i.e. the fixed effects. Our results
indicate that the spatial relationship between inventory plots
contains information that is not included in the observed
covariates, and the inclusion of the spatial random effects
improves the overall prediction (Appendix Figure A1, Fig-
ure A2, Figure A3). Compared to size classes B0 and B1,
B2 exhibited more variation, which cannot be explained by
the fixed effects. The long time lag between establishment
and inventory in the case of B2 may have contributed to
the larger share of variance that is absorbed by the random
effects in the model of B2 (Figs.5, 6). Beech regeneration
requires almost 20–40years from germination to reach a dbh
of 7cm (Hessenmöller etal. 2012). During this time, the
environmental conditions that affect growth or mortality can
change significantly. Potential seed trees can be harvested.
The lack of information on environmental conditions during
the establishment phase may lead to a greater proportion of
the variability being captured by the random effects. The
random effects give insights into unobserved processes. The
distribution of spatial random effects can be used as a sur-
rogate to derive processes from spatial patterns (McIntire
and Fajardo 2009).
In the north-western part of the example area, the occur-
rence of beech regeneration was higher than expected based
on the fixed effects, while it was lower in the eastern part,
as shown in Fig.5. The pattern suggests that a large-scale
process was responsible for this, potentially influenced by
the site. Although there was no significant effect of nutri-
ent content on regeneration occurrence, another study found
a correlation between site productivity and beech recruit-
ment (Klopcic etal. 2012). Other site variables may have a
significant impact and should be tested. In addition to site
variables, the suitability of the habitat for seed-dispersing
animals and the occurrence of other masting tree species
could also affect the probability of the occurrence of beech
regeneration and cause the pattern of random effects (Perea
etal. 2011; Löf etal. 2018).
A higher regeneration density than that predicted by the
fixed effects was found especially at the edges of the for-
est areas near the city of Dresden. The patterns appeared
on a small scale. These areas are visited for recreational
use and game may avoid these areas as a result. Mathisen
etal. (2018) found that proximity to forest roads affects the
intensity of browsing of oak as red deer (Cervus elaphus
L.) and roe deer (Capreolus capreolus L.) avoid these areas.
A lower local game density can, therefore, lead to a higher
regeneration density in the vicinity of cities. Spatial random
effects also exhibited a lower density than predicted by the
fixed effects in the interior of the forest (Fig.6). This corre-
sponded to findings of Mathisen etal. (2018) who observed
higher browsing intensity in the inner reaches of the forest.
In addition to the effect of visitors, real seed sources may not
have been included in the data. A comparison of stand inven-
tory databases and remote sensing databases indicated that
many individual seed source trees are not recorded in stand
databases (Axer and Wagner 2020). Remote sensing data
were not available for the whole study area. The small-scale
composition of the ground vegetation may also have influ-
enced the regeneration density. The most relevant task for
future research is to disentangle all that is pooled together
in random effects. Additional variables should be included
in the forest inventory for this purpose.
Conclusions
The natural regeneration and establishment of beech can
support the conversion of pure coniferous stands into
mixed stands, a task which is relevant to forest manage-
ment throughout central Europe. The spatial ZANB model,
implemented in R-INLA, provides the opportunity to analyse
regeneration data. Factors influencing the occurrence and
density of beech regeneration can be examined separately.
This improves our ecological understanding of the estab-
lishment of beech regeneration. At the same time, the study
provides information on which additional variables could
also have an influence on beech regeneration. The model
developed in this study emphasises the variability in regen-
eration across the study area. The probability of regeneration
establishing is dependent on site and overstorey character-
istics. However, the most limiting factor for regeneration is
seed availability. Due to the very limited dispersal of beech
seeds, the position of old beech trees is one of the main driv-
ers of a succession of beech trees into surrounding spruce
and pine stands. Knowledge of the location of mother trees
and of the distance to the nearest mature beech stand is,
therefore, an important source of information for manage-
ment. Based on this knowledge, natural regeneration can
be estimated spatially. Using these predictions, spatial deci-
sions on forest conversion activities can be planned at forest
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964 European Journal of Forest Research (2021) 140:947–968
1 3
management level. Areas in which artificial regeneration is
necessary can be identified. At the same time, areas in which
there is no need of planting can be identified. The evalua-
tion of the regeneration process, therefore, leads directly to
greater control over the effort that must be invested. The
model developed in this study can be replicated for other
forest districts for which sample points are available. To a
large extent, the effects determining seedling occurrence and
identified by the model are in alignment with our silvicul-
tural knowledge. The model is also relevant to practitioners,
because the variables used in the model were reported in the
context of a forest inventory.
Supplementary Information The online version contains supplemen-
tary material available at https:// doi. org/ 10. 1007/ s10342- 021- 01377-w .
Acknowledgements This work was supported by Staatsbetrieb Sach-
senforst. The authors would like to thank Sachsenforst for providing
valuable data from their regular forest inventory. We are especially
grateful to Dirk-Roger Eisenhauer for the constructive cooperation. We
thank Dominik Flatten and David Butler Manning for proofreading the
text. We would also like to thank Havard Rue and Elias Krainski for
the detailed information about their package. Our thanks also to the
anonymous reviewers for their useful suggestions.
Funding Open Access funding enabled and organized by Pro-
jekt DEAL. The work was supported financially by Staatsbetrieb
Sachsenforst.
Availability of data and materials The datasets analysed during the
study are not publicly available as the data are highly sensitive propri-
etary data of the state forest enterprise.
Declarations
Conflict of interest The authors declare that they have no conflict of
interest.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article’s Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.
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