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Methods Ecol Evol. 2021;00:1–9. wileyonlinelibrary.com/journal/mee3
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1© 2021 British Ecological Society
1 | INTRODUCTION
Tree growth is essential for forest management and forest ecosys-
tem research. Growth is the product of several abiotic and biotic
factors, interacting with a tree and its surroundings through time.
Tree growth is expressed morphologically and is vital for forest man-
agement because tree size determines timber production capabil-
ity (Monserud, 2003). In the same vein, tree grow th is the primary
driver of forest dynamics because it determines tree crown class
differentiation, mortality and regeneration, and therefore it is of piv-
otal importance for understanding how trees interact in a dynamic
ecosystem (Kimmins, 2004).
Stem analysis is a methodology for measuring tree grow th. As
pointed out by Tesch (1980), the study of tree growth is not a recent en-
deavour, and the f irst record ed observat ions of tree grow th are general ly
credited to Theophrastus (370– 285 B.C.), a student of Aristotle. Most
trees growing in temperate climates produce a distinct layer of wood
every year, indicating the particular tree's age and growth. One layer is
formed each year bet ween the bark and the previously formed wood.
This layer looks like a ring in a cross- section cut (or ‘crosscut’) of a trunk.
In stem analysis (SA), several wood discs are extrac ted along the stem
of a tree. Usually, height is measured, and the number of rings present
at each disc is counted (incomplete SA), although sometimes the radial
growth at each disc is also measured (complete SA). SA provides a record
of the height growth of a tree and its diameter growth at many points
along the stem. SA is the measuring technique that offers the most out-
standing detail on the past growth of several tree- level variables, there-
fore offering crucial useful data for tree growth studies.
Height– age pairs must be derived from the stem analysis data
to model height growth. The study of tree height growth is crucial
Received: 31 July 2020
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Accepted: 30 March 2021
DOI : 10.1111 /20 41-210X.13616
RESEARCH ARTICLE
A new algorithm for reconstructing tree height growth with
stem analysis data
Christian Salas- Eljatib1,2,3
1Centro de Modelación y Monitoreo
de Ecosistemas, Facultad de C iencias ,
Universidad Mayor, Santiago, Chile
2Vicerr ectoría de Investigació n y Postgrado,
Universidad de La Frontera, Temuco, Chile
3Departamento de Silvicultura y
Conser vación de la Natur aleza, Universidad
de Chile, Santiago, Chile
Correspondence
Christian Salas- Eljatib
Email: cseljatib@gmail.com
Funding information
Chilean Government research grants
Fondec yt No. 1191816 and FOND EF No.
ID19 10 421.
Handling Editor: Ryan Chisholm
Abstract
1. Stem analysis allows us to obtain an abundant amount of information on tree
growth. A couple of algorithms exist to utilize section height and growth ring data
for reconstructing height and age time- series information.
2. I evaluated two alternatives, a well- known and a newly proposed algorithm using
stem analysis data of four species, including deciduous and evergreen broadleaves
and a conifer. I reconstructed height– age pairs by both algorithms. I fit height
growth equations in a mixed- effects model framework for each species, using
the generated data with the respective algorithm. Comparisons considered confi-
dence intervals of the estimated parameters, as well as regression- based equiva-
lence tests.
3. Results showed that the fitted growth models obtained from both stem analysis
algorithms were statistically equivalent. However, the proposed algorithm is sim-
pler and thus provides a useful alternative to current methods.
4. Based on the findings, I recommend using this new stem analysis algorithm to
reconstruct tree height growth with stem analysis data.
KEYWORDS
equivalence testing, forest ecolog y, growth rates, height growth models, mixed- ef fects
models
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SALA S- ELJATIB
for understanding competition capabilities and its relationship with
stand structure and composition (Holste et al., 2011), as well as for
assessing forest productivity (Salas- Eljatib, 2021a). Growth can be
depicted as a series of envelopes with different taper and a decreas-
ing number of annual rings as stem height increases (Figure 1). A
count of the number of rings on a given cross- section gives the tree's
age above the section. Thus, it indicates the age of the tree at that
point. If the count is made on a cross- section at ground level, it gives
the total tree age. A section's height is attained within a period equal
to the age at the sampling time minus the section ring count. If we
could obtain a disc exactly at the point in the stem where the growth
ends for a corresponding time, we would directly know the corre-
sponding height, in this case, equal to the section height (Figure 1a).
Nevertheless, due to the trees' conical growth pattern, the actual
height at the age corresponding to the cross- section ring count will
almost always be located above the measured cross- section (Dyer
& Bailey, 1987). Let me define the cross- section age as the differ-
ence between the total number of rings at the base of a tree and the
cross- sectional number of rings. (Figure 1b). Carmean (1972) pointed
out that the height at the point of sectioning underestimates ac tual
height at the presumed age because the section will almost always
occur at some intermediate point along with the annual leader rather
than at the terminal bud itself. The actual height for the correspond-
ing age at that cross- section will be at some point above that cross-
section. The exact height or the height of the tip for a corresponding
age is known as the ‘hidden tip’ (Dyer & Bailey, 1987), as shown in
Figure 1b.
A couple of algorithms exist for reconstructing tree height–
age data pairs with stem analysis data. From the graphical one of
Graves (1906) up to the linear programming- based one of Lappi
(2006). I classify the stem analysis algorithms into two groups: (a)
‘age- based’, those that estimate the height corresponding to the
age of each cross- section, and (b) ‘height- based’, those that esti-
mate the age corresponding to the height of each cross- section.
The age- based algorithms have received the most attention in
the literature, and they are subdivided into those that use only
ring count s (e.g. Carmean, 1972; Fabbio et al., 1994; Lappi, 2006;
Lenhart, 1972; Newberry, 1991) and those that use also ring
width (e.g. Kariuki, 2002). Overall, and based on the comparisons
of Dyer and Bailey (1987), Fabbio et al. (1994), Kariuki (2002),
Lappi (2006), Machado et al. (2010), Rayner (1991) and Subedi
and Sharma (2010), Carmean's height interpolation algorithm has
performed well in almost all the published research on the topic.
Regarding the height- based algorithms, Milner (1992) proposed
one in words, but no fur ther work has been done on this t ype of
algorithm.
Dyer and Bailey (1987) pointed out that Carmean's algorithm has
two assumptions: First, annual height increment is constant for each
year for whi ch height growth is con tained within th e section. Seco nd,
the hidden tip will occur in the middle of a year's height increment.
While neither assumption is likely to be upheld, it is germane to ex-
amine an alternative approach that does not rely on either. I aim to
describe the basis of a new stem analysis algorithm and compare it
against Carmean's algorithm.
FIGURE 1 Stem analysis basis on a longitudinal split showing the progressive taper development (wood layers) for a 5- year- old tree. (a)
Horizontal dashed lines represent the ideal location of two cross- sections to accurately reconstruct the tree heights at time 1 (
t1
) and time
2 (
t2
), that is,
h
t
1
and
h
t
2
respectively. (b) In practice, the locations of cross- sections will not coincide exactly with tips. Besides, the height of
three cross- section (
h1,h2
and
h3
) and the ring count on each cross- section (
r1,r2
and
r3
) are represented. Finally, the length of the hidden tip
for
t2
is also depicted
t1t2t3t4t5
h
t5
ht4
ht3
ht2
ht1
Time
Diameter
Height
(a) (b)
t1t2t3t4t5
h
t5
ht4
ht3
ht2
ht1
Ti
me
Diameter
Height
h1 ,r1
r2
h2 ,
r3
h3 ,
Length of
hidden tip
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SALA S- ELJATIB
2 | MATERIALS AND METHODS
2.1 | Data
I used stem analysis data of sample trees selected from fixed- area
sample plots for different tree species. Data were representative
of broadleaved (two deciduous and one evergreen) and coniferous
tree species. The species were as follows: (a) Nothofagus dombeyi
(Coigüe), an evergreen tree species with a wide distribution in
Chile, being very abundant, especially in the country's southern
part. (b) Pseudotsuga menziesii (Douglas fir), a conifer native from
the Pacific Northwest of North America, extensively planted as
an exotic elsewhere. (c) Nothofagus alpina (Raulí), a deciduous
tree species that has great potential for forest management and
high timber value. (d) Nothofagus obliqua (Roble), a deciduous tree
species as well. The Nothofagus species data are obtained from
several studies described by Salas- Eljatib (2020) and cover a lati-
tudinal range bet ween 35°50′ and 41°30′S in south- central Chile
(Figure S1a). In contrast, the Douglas fir data are obtained from the
study of Monserud (1984) and cover between 42°10' and 49°50' N
in the Inland Northwest region, northern Rocky Mountains, of the
United States (Figure S1b).
All the tree- level data were collected from sample plots. For
the Nothofagus data, these plots were established in mixed- species,
even- aged, secondary stands dominated by Coigüe, Raulí or Roble.
Veblen et al. (1996) provide additional ecological features of these
forests. Meanwhile, for the Douglas fir data, the plots were estab-
lished in even- and uneven- aged stands (Monserud, 1984). For each
dataset, dominant trees were selected from fixed- area plots, pro-
vided they were healthy and of good form, of seed origin and belong-
ing to the upper canopy. The selected trees were felled, and after
measuring DBH (d) and total height (h), cross- sectional discs were
obtained at several heights along the stem. There were, on average,
10 sections per tree. Rings were counted in the laboratory. The num-
ber of plots by species is 30, 181, 53 and 62 for Coigüe, Douglas
fir, Raulí and Roble, respectively, and have between three and four
sample trees per plot.
The age of a tree is the leng th of time that has elapsed since the
germination of the seed or the sprout's budding (Husch et al., 2003).
The most common dating method for mature trees is to count the
number of annual growth rings on a transverse section or an incre-
ment core sampled at ground level, that is, at what appears to be the
root collar (DesRochers & Gagnon, 1997). I computed total age (tot.
age), at the time of sampling by linear extrapolation of ring counts
of the lowest two sections down to ground level, rounding to the
nearest integer. This age was used as the time variable for recon-
structing past height grow th. As a reference, I also computed the
breast height age (bha) as the number of rings present at 1.3 m on
the stem ( Table 1).
I tried to include more species in the analysis; however, pertinent
data from other species were unavailable. The data that could be
used have some of the following issues: small sample size; does not
have the minimum information needed, such as the number of rings
per cross- section; and trees concentrated in a single location, among
other issues. Nonetheless, obtaining data of this sort is not as easy
as a user might expect, but is the reality of most research disciplines
(Poisot et al., 2019).
2.2 | Algorithms for cross- section height–
age adjustment
Before describing the algorithms, I will introduce notation for the
raw data in a given tree. Notice that I will omit a tree's subscript to
simplify details:
hk
,
rk
and
tk
are height, number of rings and age at
the kth cross- section. For these data pairs, the Carmean algorithm
reconstructs corrected pairs
(
tk,ht
k)
and the proposed one recon-
structs
(
th
k
,hk
)
. That is to say, the main difference between the two
algorithms is either reconstructing height at a given rounded time
(denoted by
h
t
k
), or reconstructing time at a given height (represented
by
t
h
k
). From this point forward, I will refer to Carmean's algorithm as
ABA (age- based algorithm), because based on age, estimate height;
and to the proposed algorithm as HBA (height- based algorithm), be-
cause based on height, estimate time. How to compute
tk
,
h
t
k
and
t
h
k
is explained in the following sec tions.
2.2.1 | Carmean (ABA)
Carmean (1972) did not offer a mathematical expression for his algo-
rithm, but Lenhar t (1972) did. It can be expressed as follows.
where
h
t
k
is the estimated total tree height at age
tk
and
hk
is the height
of the kth cross- section. The rest of the terms were described above.
Therefore,
Δhk
is the leng th of the (k + 1)th section and
Δrk
is the dif-
ference in the number of rings bet ween the
k
and (k + 1)th sections,
provided that
r
k
>r(
k
+1)
. The age of the tree (associated with the inner-
most ring) at the kth cross- section,
tk
, is computed as
where
tb
is the reference age for height reconstruction (e.g. either
total age or breast height age). Other mathematical details of the al-
gorithm are in Supporting Information Appendix 1. In this regard, it
is important to point out that it is common practice to interpolate the
heights to every missing year between sections (using the third com-
ponent of Equation S1). This practice is unsuited because it inflates the
sample size and smooths the data before fitting a model. The lat ter's
problem is those fit statistics measuring deviations about smoothed
height increment data are misleading and strongly biased (Hasenauer
& Monserud, 1997). Hereafter Equation (1) defines the ABA, where
the length of the hidden tip is estimated as half of the periodic annual
height increment for the kth section.
(1)
h
tk=hk+
h
(k+1)
−h
k
2
[
rk−r
(
k
+1)]
=hk+Δhk
2Δrk
,
(2)
tk=tb−rk+1,
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Methods in Ecology and Evoluon
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2.2.2 | Proposed algorithm (HBA)
The HBA seeks to estimate the age of each cross- section based
on the height of that section. The idea is to reduce the calculated
cross- section age by half a year, instead of increasing the section
height by half the height increment as in Carmean's algorithm. This
approach is more straightforward than the ABA; however, no studies
have focused on its effectiveness or explained its basis. To the best
of my knowledge, the first reasoning of this kind was suggested by
Milner (1992). Therefore, in the following, I refine Milner's wording
by introducing new concepts and proposing a formal notation and
mathematical expressions.
When having stem analysis data, we observe the height of the
cross- sections. In this algorithm, I use that height as such, without
modification. Because of that, I refer to this algorithm as a height-
based one. Instead, I only alter the comput ation of the age for each
cross- section. Equation (2) allows obtaining that age for the ABA ,
symbolized by
tk
. However, given that for the proposed algorithm,
hk
is directly treated as the height of interest, we know that the
tree's age when was
hk
tall (symbolized by
t
h
k
) must be lower than
tk
.
Besides, inasmuch as the age dif ference must be bet ween 0 and 1,
I think that a uniform distribution is a suitable choice for accommo-
dating its uncertainty, as follows.
where:
t
h
k
is the age of the tree when was as tall as the height of
the kth cross- section,
uk
is a random number from a uniform distri-
bution,
u
k∼U
[
0, 1
]
,
E
is the expected value operator and the other
terms were already defined. Equation (3) is the algorithm I propose,
and hereafter refer to as ‘HBA’ (height- based algorithm). Other
mathematical details of this algorithm are in Supporting Information
Appendix 2. It is important to argue that I do not interpolate the
heights for a given year but the age for a given cross- section height
only. In summar y, although from both ABA and HBA algorithms we
reconstruct height– age data, they are built from dif ferent perspec-
tives. The struc ture of the constructed data from both algorithms is
in Table S1. Furthermore, Figure 2 illustrates both algorithms' dif-
ferences where two cross- sections are measured. In the ABA, the
estimation acts in the vertical axis (i.e. height); in contrast, the HBA
acts in the horizontal axis (i.e. age). The reconstructed height growth
series using both algorithms are represented by species, showing
a wide variety of shapes, indicating large variability in tree height
growth (Figure S2).
(3)
E[
th
k]
=tb−rk+1−E
[
uk
]
=tb−rk+
0.5,
TABLE 1 Tree sample variables' summary by species. The
variables are d is the DBH, h is the total height, tot.age is the total
age and bha is the breast height age
Species Statistic
Variable
d h tot.age bha
(cm) (m) (year) (year)
Nothofagus dombeyi (Coigüe)
(
n
= 107) Minimum 5.3 9.9 21 15
Maximum 60.2 33.7 71 68
Mean 26.6 19. 9 41.1 37.7
CV(%) 19 16.6 15.6 17. 3
Pseudotsuga menziesii (Douglas fir)
(
n
= 312) Minimum 30.5 24.2 54 51
Maximum 95.2 4 9.4 19 8 195
Mean 52.3 33.3 120.6 116.4
CV(%) 11.7 9. 6 19.9 19.8
Nothofagus alpina (Raulí)
(
n
= 169) Minimum 5.3 7.1 19 17
Maximum 49.9 31.2 81 76
Mean 24.9 20.3 45.9 42 .2
CV(%) 16. 2 15.1 17 18.4
Nothofagus obliqua (Roble)
(
n
= 155) Minimum 7.3 7. 9 14 13
Maximum 59.1 37 10 3 94
Mean 28.8 22.6 45.2 41
CV(%) 19. 4 18.3 1 9.2 19.3
FIGURE 2 Graphical representation of how the algorithms
work. Here two measurements for a given tree are represented
with black dots at heights
hk
and
hk+1
. Ages at sections
k
and
k+1
are represented by
tk
and
tk+1
, respectively, and computed as in
Equation (2). To illustrate the differences, each year (apar t from the
ones where the cross- sections are obtained) is marked with grey-
dashed ver tical lines, and the annual height increments between
each cross- sections are marked with green- dashed horizontal lines.
In the ABA (age- based, Carmean), the corresponding height at the
age
tk
, that is,
h
t
k
, is estimated, therefore the pair of reconstructed
data is
(
tk,ht
k)
, which is represented in red. In contrast, in the HBA
(height- based), the corresponding age at height
hk
, that is,
t
h
k
,
is estimated, therefore the pair of reconstructed data is
(
th
k
,hk
)
,
which is represented in blue
●
●
●
●
●
●
tktk+1
hk
hk+1
htk
thk
Age
Height
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Methods in Ecology and Evoluon
SALA S- ELJATIB
2.3 | Comparisons
To compare the algorithms' performance for correcting the section
height– age data from stem analysis, we fit the same model (i.e. base
model) using the two different datasets, one obtained with the ABA
adjustments and the other with the HBA algorithm. I do so to show
the effects of the data in the resulting fitted growth models. I repeat
the following analyses for each species.
2.3.1 | Growth law
I used the following baseline grow th func tion, which as a statistical
model is represented by.
where
hiz
is the height for the ith tree at the zth time
tiz
; while
𝛼
,
𝛽
and
𝛾
are parameters and
𝜖iz
is the random error term. Notice that
i=1, …,n
and
z=1, …,Ti
, where
Ti=Ki+1
, and
Ki
is the total number of discs
on the ith tree. Therefore,
hiTi
is the tree's total height at sampling. The
parameters have the following interpretations (Salas- Eljatib, 2020):
𝛼
is
the upper asymptote or maximum level of growth for the state variable
h
,
𝛽
is a parameter which governs the rate of change or scale parameter
and
𝛾
is a shape parameter determining the height of the growth curve
inflection point. This model is among the most widely used growth
function to study tree growth (Salas- Eljatib, 2020).
As I have several obser vations of
h
over time for the same tree,
the data have a temporal correlation. To account for this correlation,
I fit a mixed- ef fect s model. I consider the tree (in each plot) as the
random factor (i.e. group). I added random effec ts to one of the pa-
rameters of Equation (4), therefore having different model variants
(Suppor ting Information Appendix 3). The random individual effects
induce an intra- individual correlation structure that accounts for the
lack of independence among the same tree observations. I fitted
the mixed- effects models by maximum likelihood and used variance
functions to model the within- stratum errors' variance structure.
Since the Nothofagus data come from different studies, I used this
information as a stratum; meanwhile, the national forest information
was used as a stratum for the Douglas- fir data. All models were fit-
ted using the nlme package (Pinheiro et al., 2021) implemented in R
(R Core Team, 2020). All models were compared with the corrected
Akaike information criterion (AICc). From here, I obtained two mod-
els, one fitted using the ‘ABA- generated data’ and another using the
‘HBA- generated data’.
2.3.2 | Parameter estimates
To compare the algorithms' difference, I compared the parameter
estimates obtained for the same model using the generated data.
This comparison involves comparing the 95% asymptotic confidence
intervals for each parameter estimate by the type of algorithm used
to build the height growth series. In this way, I was able to examine
the sensitivity of parameter estimates to the algorithm for recon-
structing height– age data.
2.3.3 | Equivalence testing
I conducted a regression- based equivalence test within a nonpara-
metric bootstrapping framework. Traditionally, equivalence tests are
approached similarly as a traditional significant test for the same pa-
rameters, that is, one statistic is tested versus one computed from
another treatment. However, this approach is not suitable for as-
sessing the prediction per formance of models. Robinson and Froese
(2004) and Robinson et al. (2005) proposed to apply the equivalence
test in a validation context that has data (i.e. validation sample) that
are independent of the fitting process (as explained below) and as-
sessing the model in the prediction of the response variable for the
validation sample. Robinson et al. (2005) proposed a nonparamet-
ric bootstrap approach which we used in the present research. It is
essential to clarif y that I cannot directly assess equivalence among
the algorithms because the C armean one modifies the height data
while the HBA modifies the age dat a. Therefore, the t wo algorithms
neither share the same value of age nor height, on which a simple
comparison can be established. I instead compared the predictions
obtained from the same height growth model but fitted using the
generated data from each algorithm to the observed height grow th.
All the details on the application of the equivalence testing are given
in Supporting Information Appendix 4. This testing's main outcome
is to compute the proportion on which the predictions are within
the equivalent region; if it is greater than 0.95, I will reject the cor-
responding null hypothesis of dissimilarity at 5% of the significance
level. That is to say, the predic tions of the height grow th model fit-
ted with the generated data using the corresponding stem analysis
algorithm are statistically equivalent to the observed height grow th.
Equivalence testing was carried out using the equivalen ce R package
(Robinson, 2010).
2.3.4 | Bias
Apart from testing the equivalence between the algorithms, I also
compared their absolute bias. For doing so, I have proceeded using
the following four steps for each species: (a) To create simulated data
using the growth laws. I generated 1,00 0 sample trees having six
cross- sections. I predicted heights for randomly generated rounded
ages from a uniform probability density function, ranging between
10 and 200. I predicted using the baseline model and it s parameter
estimates (Table S2). Later, I added to each predicted heights a white
noise coming from a Gaussian distribution with an expected value
of 0 and a standard deviation equal to the residual model's standard
deviation, that is,
𝜎 e
. (b) To derive data of height- ring counts using
both algorithms. (c) To fit the baseline model using the derived data.
(4)
h
iz =𝛼
{
1−e−𝛽(tiz)
}1∕𝛾
+𝜖iz
,
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(d) To compute the absolute bias of both fitted models in predicting
the fundamental growth law.
3 | RESULTS
The polymorphic mixed- effect model (Equation S6) had a bet ter fit,
using as reference the AICc statistic, for both the data generated
with the ABA and the HBA, as well as for all species (Table 2). Notice
that comparisons must be only established among fitted models
using the same data because maximum likelihood- based statistics
alone are meaningless. As the polymorphic model had a better fit for
all species, it was used for subsequent analysis.
Fitting the polymorphic model (Equation S6) to height– age pairs
from the two algorithms using each species stem analysis data re-
sulted in parameter estimates significantly different from zero in
each case (Figure 3). Confidence intervals (CIs) of all the parameter
estimates of the height growth models fitted with data generated
from the two stem analysis algorithms overlapped to a large extent
(Figure 3). Overlap of CIs of all parameter estimates is almost total
for Douglas fir and Coigüe. The smaller overlap of CIs occurs for
𝛾
of
roble, however it is still greater than 80%.
The null hypothesis (
H0
) of the equivalence test is equal to the
statistics being not equivalent to the defined equivalence region.
As explained above, if the bootstrap CIs are contained within the
equivalent region, we reject
H0
; that is, we have strong evidence
against dissimilarity, favouring equivalence. The equivalence testing
results are shown in Table 3 for all species. In 100% of the boot-
strap replicates, both estimated parameters,
𝜙0
and
𝜙1
, of model (S8)
were contained within their respective equivalence regions,
I0
and
I1
respectively. Although it might seem strange to have 100%, similar
results had also been repor ted (e.g. Robinson et al., 2005), but more
than the value, what is important is to emphasize the strong evi-
dence against
H0
. The bootstrap CI for a 95% confidence level for
both parameters and all species is contained within their respective
equivalence regions. Therefore, I reject the null hypothesis of not
equivalent for both
𝜙0
and
𝜙1
, and for both all species and the two
algorithms. Predicted values obtained from both a growth model fit-
ted with data generated with the ABA and the same model but fitted
with data generated with the HBA are statistically equivalent to their
respective measured height growth values.
The average bias, an indicator of accuracy, clearly shows that
the HBA unbiasedly estimated height growth (Table 4). On the other
hand, the ABA is biased, with a slight (1%) trend towards the over-
estimation. The described pattern is consistent across all the spe-
cies. Regarding the absolute bias, the HBA has a lower value for all
species as well. These results enhance the difference bet ween the
two algorithms and favour the HBA.
4 | DISCUSSION
I have presented a pleasingly simple novel method for tree height
growth reconstruction, which involves preserving the field-
measured height data and estimating age by adding 0.5 year to the
naive age estimate obtained from ring count s. Though simple, this
new method gives results that are nearly equivalent to those of the
slightly more complex Carmean algorithm, and thus the new method
may be preferred by practitioners.
I fitted the baseline height growth model (Equation 4) in a mixed-
effects framework by allocating random ef fects to different param-
eters of the model (Equations S5 and S6). The model variant having
random effect s into the
𝛽
parameter was always the best for all spe-
cies, regardless of the algorithm used to generate the data (Table 2).
This result agrees with other studies where using the same grow th
model, the same variant, showed better goodness- of- fit indicators
(García, 1983; Hu & García, 2010). Concordantly, using a reliable
growth law for modelling the height development of the species
under study is secure. Indeed, other models can provide better
prediction capabilities. Still, I aim to use a suitable quantitative tool
that models the growth rates of the state variable height for fur ther
analyses.
Hypothesis testing is not usually taking into account when com-
paring modelling approaches. For instance, Lappi (2006) focused on
only applying a mathematical method to stem analysis data, regard-
less of the statistical implications. Here, I computed the confidence
interval of the estimated parameters by species and the algorithm
used to generate the dat a (Figure 3). Comparing the confidence in-
tervals is essential to assess the null hypothesis of both parameters
TABLE 2 Maximum likelihood- based statistic of model variants.
Each variant represents adding random effects either into the
asymptote (as in Eq. S5) or into the shape parameter (as in Equation
S6) of the baseline height growth model (Equation 4). AICc is the
corrected Akaike information criterion. Notice that comparisons
based on AICc are only valid among models using the same data.
That is to say, we can compare the AICs between the anamorphic
and polymorphic variants but only for a given data generated by a
specific algorithm
Species Model variant
Algorithm
ABA HBA
AICc AICc
Coigüe
α + ai4,191.7 4,286.0
β + bi3,954 .1 4,066.0
Douglas fir
α + ai14,590.0 14,63 8
β + bi14,071.0 14,118
Raulí
α + ai5,771.0 5,900.7
β + bi5,506.6 5 ,6 4 9.6
Roble
α + ai6,925.2 7,040. 4
β + bi6, 618.4 6,741 .1
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being equal and to see the drawn conclusion when hypothesis test-
ing is applied. In this regard, the results clearly showed overlapping
confidence intervals for the estimated parameters of both data algo-
rithm generated. Therefore, both algorithms would offer almost the
same precision on these estimated parameters. The hypothesis to be
tested using the variance components of the parameters will reach
the same conclusions.
The equivalence testing framework allowed us to per form a re-
liable assessment of t wo confronting approaches (or algorithms in
this case). The current research possess difficulties not commonly
faced when comparing competing models. The stem analysis algo-
rithms reconstruct height– age series (Figure 2; Figure S2); however,
I lack the real or obser ved height– age series, a fundamental compo-
nent of any model assessment. Inasmuch as the restriction of the
absence of actual height– age data, the equivalence testing of fers a
suitable alternative to focus on assessing whether the predictions of
the fitted models fitted from both height– age series are equivalent.
Predictions obtained from a fitted model with both algorithms' data
are statistically equivalent (Figure 3). These results are consistent
across four species' growth dat a, one coniferous and three broad-
leafs (two deciduous and one evergreen), spanning a large geograph-
ical area. The calculation of both average bias and absolute bias of
the algorithms to reproduce the growth laws (Table 4) offers a reli-
able but straightforward alternative to assess them. A gain, the HBA
outperformed the Carmean algorithm.
The HBA is more straightforward than the ABA (Carmean) and
makes fewer assumptions. ABA is the most widely used stem anal-
ysis algorithm, most likely because of the thorough assessment
conducted by Dyer and Bailey (1987). Nonetheless, this algorithm
assumes a constant periodic annual increment in height for the bolt
between two cross- sections and that the hidden tip is reached at
half of that annual increment (Figure 2). Both assumptions are barely
FIGURE 3 Asymptotic 95% confidence
intervals for the parameter estimates of
the polymorphic mixed- effects height
growth model (S6) by species, using
height– age data built from applying
the ABA (age- based, Carmean) and the
HBA (height- based) algorithms. Notice
that each column panel of the figure
represents a parameter estimate, and each
row panel represents a species
●
●
38 40 42 44 46 48 50 52
ABA HBA
Coigüe
●
●
0.010 0.015 0.020 0.025 0.030
ABA HBA
Coigüe
●
●
1.11.2 1.31.4 1.5
ABA HBA
Coigüe
●
●
38 40 42 44 46 48 50 52
ABAHBA
Douglas fir
●
●
0.010 0.015 0.020 0.025 0.030
ABAHBA
Douglas fir
●
●
1.11.2 1.31.4
1.5
ABAHBA
Douglas fir
●
●
38 40 42 44 46 48 50 52
ABAHBA
Raulí
●
●
0.010 0.015 0.020 0.025 0.030
ABAHBA
Raulí
●
●
1.11.2 1.31.4
1.5
ABAHBA
Raulí
●
●
38 40 42 44 46 48 50 52
ABA HBA
Roble
●
●
0.010 0.015 0.020 0.025 0.030
ABA HBA
Roble
●
●
1.11.2 1.31.4 1.5
ABA HBA
Roble
8
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Methods in Ecology and Evoluon
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justified, and they are unreliable. On the other hand, the HBA only
requires the measured height at a cross- section, and the age for that
section is obtained just by subtracting 0.5 (Equation 3); therefore, it
is prett y simple. If a user prefers to try the HBA stochastic version,
Equation (S3) must be used instead. Although a stochastic version
of the ABA could be built similarly, the probabilit y densit y function
(pdf) for defining the randomness's magnitude and frequency is
more challenging to justify than a uniform pdf for the HBA. Based
on the results presented here, I recommend using the HBA to recon-
struct tree height growth with stem analysis dat a. I have provided
an R code (see Salas- Eljatib, 2021b) implementing a simple example
from data of a stem analysis sample tree.
One theoretical mathematical issue of the HBA arises be-
cause it uses a non- integer time. On the contrary, in the ABA,
only integer times are considered. Therefore, the HBA mimics a
continuous- time system, concurrently the ABA mimics a discrete-
time system. Height growth occurs mainly in early spring for many
species, especially those growing in temperate zones (Kimmins,
2004). In this case, the within- year growth dynamics would be
flat most of the year. I cannot define a single time associated with
height (i.e. height did not change during most of the year). The
HBA, however, does not attempt to represent the seasonal pat-
tern of growth. Despite the baseline height growth model being
a solution to a differential equation, in applications, I restrict it to
integer time values. Leary (1985) advocated for similar uses of dif-
ferential equation in forest modelling. In practice, for computation
purposes, some available alternatives use corrections depending
upon the time within the year that we measure the state variable.
Other options are available; regardless, they are well beyond the
scope of the present paper.
5 | CONCLUSIONS
The stem analysis algorithm presented here increases the flexibility
of using this technique for reconstructing time- series data on height
growth. This type of data is essential to many areas of ecology and
sustainable forest management, and improving their analysis is un-
doubtedly useful. The two algorithms are presented and compared
using modern and suitable statistical models. The proposed algo-
rithm is simpler, based on fewer assumptions, more straightforward
than the classical one, and finally yields equivalent results. I finally
recommend its use.
ACKNOWLEDGEMENTS
The author thanks Drs Robert Monserud, Pablo Donoso, Hans
Grosse, Alicia Ortega, and Pablo Gajardo and Prof. Patricio Núñez
for providing the data used here. He especially thanks Dr Oscar
García for discussing aspect s that he have tried to organize here. The
reviewers and the Associate Editor greatly improved earlier versions
of the manuscript by providing valuable comments.
Species
𝝓0
𝝓1
I0
Bootstrap CI
I1
Bootstrap CI
Lower Upper Lower Upper Lower Upper Lower Upper
Coigüe
ABA 7.7 1 12.86 10.78 11. 38 0.75 1.25 0.97 1.07
HBA 7.52 12.54 10.49 11 .09 0.75 1.25 0.97 1.06
Douglas fir
ABA 15.37 25.62 19. 2 5 19.7 7 0.75 1.25 0.87 0.92
HBA 15.27 25.45 19. 12 19.63 0 .75 1.25 0.88 0.91
Raulí
ABA 7.54 12.56 9. 54 9.98 0.75 1.25 0.936 1.00
HBA 7.35 12.25 9.28 9. 72 0 .75 1.25 0.94 3 1.03
Roble
ABA 8. 51 14.18 10.27 10.78 0.75 1.25 0.86 0.95
HBA 8.28 13. 80 9.96 10.46 0.75 1.25 0. 87 0.94
TABLE 3 Summary of equivalence-
based regression results by species. The
equivalence region (ER) for the intercept
𝜙0
, termed
I0
, was set to the mean
measured height ±10%. The ER for the
slope
𝜙1
, termed
I1
, was set to 1 ± 0.25.
The 95% bootstrap- based confidence
intervals (CI) were computed for both
𝜙0
and
𝜙1
. Notice that the proportion
of times that each of these parameter
estimates were contained within the
respective ER was 1 for all the cases
TABLE 4 Mean bias and absolute bias of the stem analysis
algorithms by species
Species Algorithm Bias (%)
Absolute
bias (%)
Coigüe ABA −0.9063 9.0 8 31
HBA −0.0105 9.053 5
Douglas fir ABA −1. 2 751 15.14 43
HBA 0.0217 15.0737
Raulí ABA −0.9506 8 .1791
HBA −0.0085 8.140 4
Roble ABA −0 .9783 8.8320
HBA −0.0181 8.7921
|
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Methods in Ecology and Evoluon
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PEER REVIEW
The peer review history for this article is available at https://publo ns.
com/publo n/10 .1111/2 041- 210X.13616.
DATA AVAIL ABI LIT Y S TATEM ENT
Data and an R code implementing the proposed algorithm are depos-
ited in the Dryad Digital Repository https://doi.org/10.5061/dryad.
qnk98 sfgc (Salas- Eljatib, 2021b).
ORCID
Christian Salas- Eljatib https://orcid.org/0000-0002-8468-0829
REFERENCES
Carmean, W. H. (1972). Site index curves for upland oaks in the central
states. Forest Science, 18, 10 9– 12 0 .
DesRochers, A ., & Gagnon, R. (1997). Is ring count at ground level a good
estimation of black spruce age? Canadian Journal of Forest Research,
27, 126 3– 1267.
Dyer, M. E., & Bailey, R. L. (1987). A tes t of six methods for es timating
true heights from stem analysis data. Forest Science, 33, 3– 13 .
Fabbio, G., Frattegiani, M., & Manetti, M. C. (1994). Height estima-
tion in stem analysis using second differences. Forest Science, 40,
329– 3 40 .
García, O. (1983). A stochastic differential equation model for the height
growth of forest stands. Biometrics, 39, 1059– 1072.
Graves, H. S. (1906). Forest mensuration (p. 458). John Willey & Sons.
Hasenauer, H., & Monserud, R. A. (1997). Biased predic tions for tree
height increment models developed from smoothed ‘data’. Ecological
Modelling, 98, 13– 22. ht tps://doi.org/10.1016/S0304 - 380 0
(96)01933 - 3
Holste, E. K., Kobe, R. K., & Vriesendorp, C. F. (2011). Seedling grow th
responses to soil re sources in the understor y of a wet tropical forest.
Ecology, 92, 1828– 1838. https://doi.o rg/10.1890/10- 1697.1
Hu, Z., & García, O. (2010). A height- grow th and site- index model for
interior spruce in the Sub- Boreal Spruce biogeoclimatic zone of
British Columbia. Canadian Journal of Forest Research, 40, 1175– 1183.
htt ps://doi.org /10.1139/X10- 075
Husch, B., Beers, T. W., & Kershaw, J. A. (2003). Forest mensuration (4th
ed., p. 443). Wiley.
Kariuki, M. (20 02). Height estimation in complete stem analysis using
annual radial grow th measurements. Forestry, 75, 63 – 74. ht tps ://do i.
org/10.1093/fores try/75.1.63
Kimmins, J. P. (2004). Forest ecolog y (3rd ed., p. 611). Pearson Prentice
Hall.
Lappi, J. (2006). Smooth height/age cur ves from stem analysis with linear
programming. Silva Fennica, 40, 291– 301. https://doi.org/10.14214/
sf.344
Leary, R. A. (1985). Interaction theor y in forest ecology and management (p.
219). Martinus Nijhoff/Dr. W. Junk Publisher s.
Lenhar t, J. D. (1972). An alternative procedure for improving height/age
data from stem analysis. Forest Science, 18, 332.
Machado, S. A., Rodrigues, L. C ., Figura, M. A., Teo, S. J., & Mendes, R. G .
(2010). Comparison of methods for estimating height from complete
stem analysis data for Pinus taeda. Ciência Florestal, 20, 45– 55.
Milner, K. S. (1992). Site index and height growth cur ves for ponder-
osa pine, western larch, lodgepole pine, and Douglar- f ir in western
Montana. Western Jour nal of Applied Forestry, 7, 9– 14 .
Monserud, R. A . (1984). Height growth and site index curves for Inland
Douglas- fir based on stem analysis data and forest habitat type.
Forest Science, 30, 94 3– 965 .
Monserud, R. A . (2003). Modeling stand growth and management. In R.
A. Monserud, R . W. Haynes, & A . C. Johnson (Eds.), Compatible forest
management (pp. 145– 175). Kluwer Academic Publishers.
Newberry, J. D. (1991). A note on Carmean's estimate of height from
stem analysis data. Forest Science, 37, 368– 369.
Pinheir o, J., Bates, D., DebRoy, S., S arkar, D., & R Core Team. (2021). nlme:
Linear and nonlin ear mixed effects models. R package version 3.1- 152.
Poisot, T., Bruneau, A ., Gonzalez, A ., Gravel, D., & Peres- Neto, P. (2019).
Ecological data should not be so hard to find and reuse. Tre nds
in Ecology & Evolution, 34, 494– 496. https://doi.org/10.1016/j.
tree.2019.04.005
R Core Team. (2020). R: A language and environment for statistical comput-
ing. R Foundation for Statistical Computing. Retrieved from http://
www.R- proje ct.org
Rayner, M. E. (1991). Estimation of true height from karri (Eucalyptus di-
versicolor) stem analysis. Australian Fores try, 54, 105– 108.
Robinson, A. (2010). equivalence: Provides tests and graphic s for assessing
tests of equivalence. R package version 0.5.6.
Robinson, A. P., Duursma, R. A ., & Marshall, J. D. (20 05). A regression-
based equivalence test for model validation: S hifting the burden of
proof. Tree Physiology, 25, 903– 913. https://doi.org/10.1093/treep
hys/25.7.903
Robinso n, A. P., & Froese, R. E. (2 004). Mode l validation usi ng equivalen ce
tests. Ecological Modelling, 176, 349– 358. ht tps://doi.org/10.1016/
j.ecolm odel.2004.01.013
Salas- Eljatib, C. (2020). Height growth– rate at a given height: A mathe-
matical perspective for forest productivity. Ecological Modelling, 431,
109198. https://doi.org/10.1016/j.ecolm odel.2020.109198
Salas- Eljatib, C . (2021a). An approach to quantif y climate- productivity
relationships: An example from a widespread Nothofagus forest.
Ecological Applications. https://doi.org/10.1002/eap.2285
Salas- Eljatib, C. (2021b). Data from: A new algorithm for reconstructing
tree height growth with stem analysis data. Dr yad Digital Repositor y,
https://doi.org/10.5061/dryad.qnk98 sfgc
Subedi, N., & Shar ma, M. (2010). Evaluating height- age determination
methods for jack pine and black spruce plantations using s tem anal-
ysis data. Northern Journal of Applied Forestry, 27, 50– 55. https://doi.
org /10.1093/njaf/27.2 .50
Tesch, S. D. (1980). The evolution of forest yield deter mination and site
classification. Forest Ecology and Management, 3, 169– 182. https://
doi.org/10.1016/0378- 1127(80)90014 - 6
Veblen, T. T., Hill, R. S ., & Read, J. (Eds.). (1996). The ecolog y and biogeogra-
phy of nothofagus forests (p. 428). Yale University Press.
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How to cite this article: Salas- Eljatib C. A new algorithm for
reconstructing tree height growth with stem analysis dat a.
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