Planting urban trees is substantially important for the future development of urban spaces. The foliage reflects, scatters, and absorbs incoming shortwave solar radiation, thus reducing the energy flow to the underlying surface, including streets, houses, and humans (Mehra, 2021). Shaded areas, with primarily anthropogenic substrates, are not as heated as non-shaded areas; subsequently, the temperature radiation and its influence on morbidity are comparatively lower in shaded areas. An increase of even 1°C in the perceived temperature in summer causes a 3.12% increase in mortality (95% CI: 0.60%–5.72%) (Jahn, et al., 2013). Shortwave solar radiation is one influencing effect on the urban environment. Energy-rich wavelengths between 400 and 900 nm (visible light) can be manipulated with high tree density. The reduction of these wavelength in an oak wood for the temperate zone at 1 pm in spring and summer was 0% or 55–88% for the upper crown space (light crown), 10–25% or 95–98% for the lower crown space (shade crown), 20–55% or 97.5–99.25% for the shrub layer, and 20–55% or 98.5–99.5% for the herb layer as well (Kalusche, 2016). In addition to shading, which reduces direct solar radiation, plant transpiration can also have a cooling effect (Bongardt, 2006; Kuttler, 2010b; Moser, et al., 2017).

Plant species play an essential role in urban biodiversity and water balance. Necessary regulations (e.g., pest control by birds and bats and pollination by flies, beetles, bees, and butterflies) require balanced biodiversity, in which species sometimes depend on urban trees as a habitat. Urban trees have diverse ecological niches. Therefore, the importance of urban biodiversity cannot be overestimated (Gloor & Göldi Hofbauer, 2018).

In addition to providing biomass for district heating supply, the output of urban trees includes food and medical products. The flowers of Tilia cordata and T. platyphyllos have a wide range of medicinal applications and are considered medicinal plant components for gastritis (Riffel, 2021). Plant components of Crataegus monogyna and C. laevigata are used as treatments for the cardiovascular system (Riffel, 2021). Additionally, food production from fruit trees is an essential factor for sustainable food security.

The filtering functions of green areas have often been investigated and described (Baldacchini, et al., 2019; Wróblewska & Jeong, 2021). An Acer platanoides has a height of 9 m, a total leaf count of 41,000, an average leaf area of 68 cm², a total leaf area of 278 m², and a total dust layer of 2 kg during the growing season (of which 20% is particulate matter), while 3.5 kg of particulate matter is released into the environment during the same period (Breuste, et al., 2016).

Although landscape structures and associated biodiversity can promote human well-being (Kardan, et al., 2015; Moser, et al., 2017; Heise & Hallermayr, 2022), they also cause disadvantages (Bergmann & Straff, 2015; Moser, et al., 2017; Breuste, 2022). Trees, in particular, are essential for emotional sensation, as they can accompany people from childhood (e.g., apple trees in the garden) (Gebhard, 2020). A tree evokes positive emotions in humans, particularly through an intact, large crown with rich greenery (Heise & Hallermayr, 2022). Street trees have a predominantly positive emotional effect on people and can reduce stress (Flade, 2018). This stress reduction or recovery effect reduces susceptibility to accidents during road transport (Flade, 2018). In addition to the cultural and regulatory services that urban trees provide to ecosystems, they also provide services related to the allometric conditions of the individual tree species. Accordingly, functions for biomass, age determination, height growth, diameter growth, and trunk and crown shape have been created in different parts of the world (cf. Snorrason & Einarsson, 2006; Lukaszkiewicz & Kosmala, 2008; McPherson, et al., 2016; Hayat, et al., 2017; Moser, et al., 2017; Riedel, et al., 2017; Riedel & Kändler, 2017; Lei, et al., 2018; Lockow, 2022). Knowledge regarding primary growth conditions is crucial to determine the impact of tree species on human wellbeing. For example, the contribution of trees to urban ecosystem services often depends on tree dimensions. The ecosystem services of urban green infrastructure must be integrated into the general urban planning structure to ensure the sustainable development of urban spaces.

This study focuses on the basic theory for modeling the growth of breast height diameter (bhd), height, and crown diameter as a basis for estimating the ecosystem services of trees in cities as part of urban green infrastructure for municipal databases. The study area was the city of Magdeburg in Saxony-Anhalt, Germany. This study focused on identifying the factors that influence different allometric parameters. In the future, the performance of tree species in various urban ecosystems will be integrated into municipal planning tools.

Magdeburg, located in the heart of Germany, has a continental climate. This area is part of the Central German Dry Zone. The regional climate data were obtained from the Climate Data Center of the German Weather Service for weather station number 3126 (Magdeburg) [https://opendata.dwd.de/]. The folder contains the recorded values, explanations of the measurement technology used, and the identification of incorrect values. Temperature was defined as the daily average, maximum, and minimum at 2 m height. Temperature recordings were available for the period from 1 January 1881 to 31 December 2021. During this period, 731 records are missing, two of which were outside the time series under consideration, 30 between 1 and 31 April 1940. The remaining 699 missing values were recorded between 1 February 1945 and 31 December 1946 (WWII). Precipitation data were also used as influencing variables. Precipitation data were available for the period between 1 January 1881 and 31 December 2021. A total of 715 missing data records were observed, 14 of which were outside the observation period. Most shortfalls (700) were observed between 31 January 1945 and 31 December 1946. The last abiotic influencing variable, which was derived from the climate data, is the irradiation intensity. Sunshine duration, as a representative quantity, was recorded only from 1 October 1934. The years between 1934 and 31 December 1935 are marked by frequent data gaps. A total of 412 missing data values are recorded for the period before 1936. In March 1941, there were six data gaps. The period between 1 February 1945 and 31 March 1951 was characterized by total failure (2,250 values). The climate data gaps were filled through estimations based on the expected value.

The mean groundwater levels were obtained from the data on the Water Resources Services of the State Office for Flood Protection and Water Management (LHW) [https://gld.lhw-sachsen-anhalt.de/]. The groundwater gauges provide data for a time series of regular measurements between early 1960 and 2022. The mean groundwater level was represented as cm below ground level.

The infiltration coefficients were derived from the database for the soil property maps of the state of Saxony-Anhalt (©State Office for Geology and Mining Saxony-Anhalt) [available at: https://metaver.de/trefferanzeige?docuuid=C3444D0C-5AD7-4312-8A77-DB9B746940E3].

Range diagnoses were used to determine the relationship between individual growth rates. Plant-specific range diagnoses according to Jäger (2017) and Müller et al. (2021) were interpreted as described in Thomas (2018). For the temperature gradient, the flora zones and elevation stages were used. The elevation levels were interpreted according to Breckle and Rafiqpoor (2019) and Thomas (2018). Accordingly, it was observed a temperature decrease of 0.55 K and a precipitation increase of 100 mm per 100 m elevation increase.

The Magdeburg tree cadaster was used to derive the allometric basis to estimate the ecosystem services of tree species. As of June 2022, the tree cadaster Magdeburg included 89,766 individual trees. The tree species and genera Acer campestre, A. platanoides, Malus spp., and Quercus robur were used in this investigation. The tree cadaster contains position information, planting years, and information on the most crucial tree parameters, including bhd (or trunk circumference), height, and crown diameter. The tree species under investigation comprise about 23.2% (20,712 trees) of the total number of urban trees in Magdeburg on public green spaces that are registered in the tree cadaster. Of these 20,712 trees, 3,892 could be used to derive the allometric function. Most had no age information, which is why no corresponding correlations could be identified.

Acer campestre is widespread in Europe, tolerant of environmental stress, and prefers warm, sunny, nutrient-rich, and calcareous locations (StMELF, 2022b). According to Gloor and Göldi Hofbauer (2018), the biodiversity index (BD-I) is 4.0. As a field shrub, it is wind-tolerant and has an extensive cardiac root system (StMELF, 2022b). The ecological amplitude holds the average temperature and the precipitation for the geographical distribution. A. cam is located according to the plant geography continental stages (PGCS) between C1 and C6 for the European continent (Jäger, 2017). The annual mean temperature amplitude of this PGCS ranged between 0°C and 20°C, according to the mean annual temperature for Europe between 1991 and 2010. The precipitation in that area lies between 300 and 1800 mm per year, with a main distribution of <1,300 mm precipitation per year. According to Ellenberger, the plant-specific indicator value for the light number was five (for young trees), the temperature number was six, and the humidity number (soil) was five (Jäger, 2017). The yellow-to-orange autumn-colored leaves contrast with the often-gray urban infrastructure. Depending on its location and growth habit, A. campestre can reach standard heights of 1–5 m as a shrub and up to 20 m as a solitary tree (Jäger, 2017). The flowers are greenish-yellow and inconspicuous. Pollination during the flowering period from May to June is performed by wind and insects (Jäger, 2017). Its seeds are short-lived cold germs spread by the wind (Jäger, 2017). A total of 1,041 A. campestre were identified for further investigation.

Acer platanoides is a penumbra tree species but also tolerates full sun (StMELF, 2022c). The easily decomposable organic litter (StMELF, 2022c) can be used extensively in parks for humus buildup. According to Gloor and Göldi Hofbauer (2018), the BD-I is ∼3.6. Its yellowish-green flowers serve as a food source for insects from April to May. Its average height is 25 m, and it can be found as forest and street trees in Germany (Jäger, 2017). It is robust against industrial stress and is often used in public greenery (StMELF, 2022c). It has a deep, compact root system that can reach lower soil layers (Georg August University Göttingen, Germany, n.d). The PGCS is between C2 and C5 for the European continent (Jäger, 2017). The mean annual temperature ranges from 0°C to 15°C, and the annual precipitation ranges from 600 to a regional maximum of 1800 mm. It can also be used to produce furniture (StMELF, 2022c). A total of 1,140 A. platanoides specimens were included in this study.

Malus spp. belong to the rose family (Rosaceae), which is diverse due to human selection, the feralization of cultivated forms, and bastard formation (StMELF, 2022a). Due to its floral splendor, it is a vital insect food plant. In addition, fruits serve as a food source. According to Gloor and Göldi Hofbauer (2018), the BD-I was 4.2. Malus spp. and are 2–10 m high, depending on the species. The flowering period is from April to May (Jäger, 2017). It is a shallow rooter often used in landscapes as hedges and for mixed planting (StMELF, 2022a). The PGCS for Malus domistica is between C5–C7 for the west Asian continent; otherwise, Malus sylvestris ranges from C1 to C7. The annual precipitation for that area lies between 300–1800 mm per year. A total of 540 M. spp. exemplars were used in this study.

Quercus robur is widespread in Germany (StMELF, 2022d) but has a decreasing trend (Jäger, 2017). Q. robur tolerates waterlogging but not groundwater stress (StMELF, 2022d). Trees represent diverse ecological niches and are valuable for biodiversity. The BD-I is 5.0 out of the 5.0 possible scale points (Gloor & Göldi Hofbauer, 2018). Q. robur has a deep taproot and (StMELF, 2022d) grows to a height of 40 m (Jäger, 2017). Q. robur blooms in May. A total of 1,171 individual Quercus robur trees were considered in this study.

First, the factors influencing the allometric conditions were examined. Multiple regression identified an essential relationship between the immediate environment, climate conditions, and specific growth. The main model concept for the simulation of urban climate conditions and the coupled process with the urban green infrastructure is shown in the following Figure 1.

According to our null hypothesis, climatic conditions and the surrounding environment do not influence the allometric growth conditions (Equation 34). Therefore, the null hypothesis can be discarded if an effect is registered in the multiple regression analysis.

Environmental effects should be examined for species-specific ecological amplitude. For this purpose, maximum daily values of 17°C–19°C, 19°C–21°C, and 21°C–23°C and annual optimum precipitation ranges of 450 mm–550 mm, 550 mm–650 mm, and 650 mm–750 mm were investigated as the optimal temperature conditions. The combinations of the respective amplitude resulted in nine scenarios. In scenarios 1–3, the annual precipitation (optimum) was 450 mm–550 mm with changing temperature optima from 17°C–19°C and 21°C–23°C. Scenarios 4–6 investigated an annual precipitation optimum between 550 mm–650 mm with the likewise changing temperature optima as scenarios 1–3. In scenarios 7–9, the optimum annual precipitation was set as an ecological amplitude between 650 mm–750 mm, and all temperature optima were considered.

The following multiple regression model was used to identify the influencing factors. Breast height diameter ( b h d ), tree height ( h ), and crown diameter ( c d ) were measured (Equation 36). The data preparation was carried out as described as follows. To complete the deficits, the immediately preceding and subsequent periods were used. The selected periods estimated according to further use used the following equation. x ¯ j = ∑ i = 1 m x j , m n j , m . (1)

The mean error of the expected value was determined as follows: s x , j = ∑ i = 1 m x j , m − x ¯ j 2 n j , m − 1 . (2)

For further processing, a normally distributed random variable was used for the periods concerned and the expected climatic conditions. To do so, the 95% confidence interval was interpreted as follows: s x , 95 = 1.96 ∗ s x , j . (3)

The final estimate was made in the interval: x ^ j = x ¯ j ± s x , 95 . (4)

Figure 2 should be consulted for a first impression of the data completion. In the following, the daily maximum temperature was used to derive the energy production and the lower activity during physiological stress. Photosynthesis as an energy supplier for the tree is dependent on the ambient temperature and solar radiation. If this ambient temperature is outside a plant-specific temperature level T o , l − u , a decrease in photosynthetic rate is assumed, resulting in a corresponding decrease in the availability of resources that support growth. The influence of temperature was considered as follows: f T , i = e T − T o , l 5   f o r   T < T o , l , e T o , u − T 5   f o r   T > T o , u . (5)

Figure 3 shows the ranges of the examined relative plant-physiological optima. Next, the daily f T , i ratings over the vegetation periods ( n v p ; i ) were summed over the years after planting. F T = ∑ i = p l a n t   y e a r 2021 f T , i . (6)

The absolute daily ratings were related to the number of vegetation periods experienced at the theoretical optimum. The researched tree species were in zonobiome VI. The vegetation period was interpreted as the moving average ( n v p ¯ ) for the last 14 days. The mean daily air temperature was used as an indicator to initialize the growth period (e.g., leaf sprouting or flowering). The following equation and condition were used: n v p ; i ¯ = T m e a n , i + n v p ; i − 1 ¯ ∗ 13 14 , (7)

where T m e a n , i is the temperature for the day i and the moving average weighted for the last 13 days ( n v p ; i − 1 ¯ * 13 ). The growing season was ultimately considered as follows: n v p ; i = n v p ; i ¯ ≥ 10 ° C = 1 , n v p ; i ¯ < 10 ° C = 0 . (8)

The growing season for the year j was calculated as follows: n v p ; j = ∑ i = d a y   o f   y e a r   j 365 / 366 n v p ; i . (9)

Figure 4 shows the calculated growing season. The summed days of the growing season since the tree was planted were calculated using the following equation: N v p = ∑ j = p l a n t   y e a r 2021 n v p ; j . (10)

The relative growth conditions in terms of local temperature gradients were determined as follows: T o , r = F T N v p . (11)

The precipitation was summed for the respective year and the corresponding vegetation period. This rainfall donation was distributed to a site-specific infiltration coefficient. The data were retrieved from the WFS download service using the Quantum GIS software and accessed the expansion of the city of Magdeburg in a tailored way. The calculation of the effective rainfall followed a simple approach based on the infiltration capacity of the respective location of the tree individual. The data from the LAGB were interpreted as follows: ψ i = C N L A G B 100 , (12) w h e r e   C N L A G B represents the surface runoff for unvegetated ground. No C N -values stored in this dataset were for urban areas. At this point, the trees were assigned different C N -values. For the trees located on roads, C N = 93 , while for parks or playgrounds, C N = 83 . For precipitation, the annual amounts were considered so that n y represented the number of years after planting ( f P , i = 1 ). For the surface runoff of the optimal constellation ( ψ o ), the CN was set to 60 and an overall pessimistic approach was taken. The factor for the rain donation was simplified according to N e = 1   f o r   ∑ i = p y 2021 d i ∗ f c r − ∑ i = p y 2021 E T 0 ; i ≥ 0 , ∑ i = p y 2021 f P , i ∗ 1 − ψ i n y ∗ 1 − ψ o + ∑ i = p y 2021 d i ∗ f c r ∑ i = p y 2021 E T 0 ; i   f o r < 0 . (13)

The relative precipitation optimum was considered as follows: f P , i = e P − P o , l 50   f o r   P < P o , l , e P o , u − P 50   f o r   P > P o , u . (14)

Figure 5 shows the relative optimum in terms of annual precipitation. Next, the daily f T , i ratings over the vegetation periods ( n v p ; i ) were summed up over the years after planting. F P = ∑ i = p l a n t   y e a r 2021 f P , i . (15)

The next influencing factor in the water cycle is the uptake of groundwater ( f c r ). Capillary rise provides information on the potential for daily replenishment of groundwater. In order to relate the daily replenishment to the daily evaporation, the days (d_i ) of the respective year must be taken into account. To assess groundwater uptake, data from groundwater measuring stations of the State Office for Flood Protection and Water Management Saxony-Anhalt [https://gld.lhw-sachsen-anhalt.de/] were used and transferred to QGIS. The mean groundwater levels were set in relation to the upper edge of the terrain at the measuring point. This generated a factor for the further estimation of the groundwater distances to the top of the terrain. These factors have now been applied to the digital terrain models (DGM) of the State Office for Surveying and Geoinformation Saxony-Anhalt [https://www.lvermgeo.sachsen-anhalt.de]. Both data sets were prepared by multiplication in the QGIS grid computer for the estimation of the groundwater distance in the study area. Subsequently, an approximate groundwater distance based on the long-term mean value for the individual tree could be assigned to each tree through a statistical analysis of the generated raster layer. For each tree species, different root depths were assumed based on the species-specific ecological amplitude, which was related to the mean groundwater level and soil system including ground density storage and capillary rising potential. The importance of the accessibility of the groundwater aquifer was considered as follows: G W r = G W l ; m e a n − D r o o t . (16)

It should be noted that at this stage, the previous function does not consider the influence of changing groundwater levels but rather the rootability of the soil. D r o o t = f r s ∗ S L . (17)

This takes into account the factor for soil-dependent rootability (f_rs ) and plant-specific rooting depth (S_L ). The soil compartment has a great influence on rootability. Accordingly, the average values from the official Soil Mapping Guidance (Sponagel, Grottenthaler, Hartmann et al., 2005) were used for the different soil types. A classification of the storage density of the soil was made to be able to estimate the air capacity and, also, the rooting depth accordingly. The storage density for parks and playgrounds was set to category 2, while that for street trees was set to category 4.

The reference evapotranspiration was estimated using the Hargreaves equation (Allen, Pereira, Raes et al., 1998). E T 0 ; i = 0.0023 ∗ T m e a n ; i + 17.8 ∗ T max ⁡ ; i − T min ⁡ ; i ∗ 0.408 ∗ R A ; k . (18)

Herein, the temperature was represented as the daily (index i) min, mean, and max at 2 m elevation above the ground in °C. The extraterrestrial solar radiation R A for a month k is given in Allen et al. (1998).

The influence of tree growth competition was determined by the overlap of the crown projection area, assuming a 1:1 ratio between the crown and root space. This means that the projection surface corresponded to the area of the crown expansion. However, a uniform crown expansion was assumed. This is usually not the case as individual branches compete for light resources. This competitive situation affects the growing conditions. Depending on the ecological preference for lighting conditions, which can be estimated by the tree-specific indicator values, the growth and, thus, the pursuit of solar energy differ. According to the type of competition, it can be divided into different states, such as a one-sided advantage through the use of resources and mutual benefit (for example, shade plants in the shade of a light plant). For the next steps, each TOI (tree of interest) was assigned a position in the center of a coordinate system with the respective coordinate origin in the point P T O I , i λ T O I , i ; φ T O I , i . Quadrants I–IV were used for the projection of the azimuth and maximum elevation angles for shading by the competitor. The position of the competitor in metric coordinates was determined for the horizontal coordinate point as follows: ∆ λ m = λ K − λ T O I (19) and for the vertical coordinate point as follows: ∆ φ m = φ K − φ T O I . (20)

The distance from the TOI to the potential competitor was determined as follows: D K , i = ∆ λ m 2 − ∆ φ m 2 . (21)

Figure 6 shows how the position of the competitors affects the direct irradiation of the TOI in the context of the azimuth angle and the path of the sun (horizontal). The determined point coordinates of the respective competitor were used to calculate the azimuth angles using the following equation: α K , i = 90 − arctan ∆ φ m ∆ λ m   f o r   ∆ λ m > 0 ; ∆ φ m > 0 , 90 ° + arctan ∆ φ m ∆ λ m   f o r   ∆ λ m > 0 ; ∆ φ m ≤ 0 , 270 ° − arctan ∆ φ m ∆ λ m f o r   ∆ λ m ≤ 0 ; ∆ φ m ≤ 0 , 270 ° + arctan ∆ φ m ∆ λ m f o r   ∆ λ m ≤ 0 ; ∆ φ m > 0 . (22)

The shading by the competitor, significantly influenced by the elevation angle of the two trees to each other, was determined using the tree heights and the distance as follows: γ K , i = arctan h K − 3 ∗ h T O I 4 D K . (23)

Finally, considering the azimuth angle and elevation angle, the effect of shading was determined. For this purpose, the course of the sun was set according to the definition of DIN5034 from Quaschning (2013).

A simple approach provides the approximate period during the day that the tree is shaded. For this purpose, the crown radius of the competitor was used to determine the minimum and maximum X and Y extent in the aforementioned coordinate system. The new equation for the minimum and maximum Y distance (North-South-Point coordinates) based on Equation 20 is shown as follows: ∆ φ m , m i − m a = φ K − c d 2 − φ T O I   f o r   m i n i m u m , φ K + c d 2 − φ T O I   f o r   m a x i m u m . (24)

The further equation for the minimum and maximum X distances (West-East-Point coordinates) based on Equation 19 is shown as follows: ∆ λ m , m i − m a = λ K − c d 2 − λ T O I   f o r   m i n i m u m , λ K + c d 2 − λ T O I   f o r   m a x i m u m . (25)

Next, the respective new point coordinates were converted into the corresponding azimuth angles by inserting them into Equation 22. From this followed the recalculation of the minimum and maximum azimuth angles in the north-south direction an exchange of the term ∆ φ m with ∆ φ m , m i − m a and for the recalculation of the east-west extension an exchange of the term ∆ λ m with ∆ λ m , m i − m a . The azimuth angles were plotted in the four directions to provide an estimate of the shading of the central tree. The difference between the maximum and minimum angles is defined as the daily shading. The sunrise and sunset angles were used to preselect the potential shading hours. For this purpose, the azimuth angles of the north-south extension were considered more closely. If these were outside 60°–305°, no shading effect was considered. Finally, temporal shading hours were determined as follows: S h t , d = max ⁡ ⁡ [ α K , i , m a ∆ λ m , m a ∆ φ m , m a − min ⁡ ⁡ [ α K , i , m i ∆ λ m , m i ∆ φ m , m i 15 . (26)

As shown in Figure 7, a daily or seasonally fluctuating disturbance impulse was associated with the numbers, height, crown dimension, and species of the competitors. For example, birch (e.g., Betula pendula) species have a lower leaf area than a same-sized oak (e.g., Quercus robur). Thus, only individuals who cut the path of the sun had influenced illuminance. The simplified hours of shading were set as follows to the sun elevation in the growing season: f S D , i = Y a + Y b ∗ S h t , d   i f   60 ° < α K ; m a ≤ 90 ° ; γ K > 15 ° , Y c ∗ S h t , d   i f   90 ° < α K ; m a ≤ 110 ° ; γ K > 20 ° , Y c ∗ S h t , d   i f   110 ° < α K ; m a <   255 ° ; 20 ° < γ K ≤ 42 ° , Y a + Y b + Y c ∗ S h t , d   i f   110 ° < α K ; m a ≤ 255 ° ; 42 ° < γ K ≤ 60 ° , Y c ∗ S h t , d   i f   255 ° < α K ≤ 280 ° ; γ K > 20 ° , Y a + Y b ∗ S h t , d   i f   280 ° < α K ≤ 305 ° ; γ K > 15 ° , 0   f o r   a l l   o t h e r . (27)

The day of the year was set in the previous function as follows. Day 91 is April 1, while day 172 is the summer solstice (June 21). For the first period of the growing season, we used the dates described earlier. After June 21, the sun elevation decreases until the end of the growing season (day 304). Day 243 is August 31, during which the sun elevation is the same as that on April 1. The limits defined by the respective days in the year are provided with the following parameters: Y a = 172 − 91 ;   Y b = 243 − 172 ;   Y c = 304 − 243 .

The reduction of the solar radiation from shading by a competitor is summed for the TOI, as given in the following equation: f S D = ∑ i = 1 n f S D , i . (28)

The reduction of solar radiation is considered as follows: S D K = S D r I − f τ ∗ f S D ∑ i = 1 n S D d l , i . (29)

The incoming relativized solar radiation S D r I was determined with sunshine hours S D d s , i per vegetation period and year, as well as the average daylight hours S D d l , i , whereby the basic ratio was assumed: S D r I = ∑ i = 1 n S D d s , i ∑ i = 1 n S D d l , i . (30)

For the rest of the procedures, the S D d l , i FAO default values for locations with a latitude of 50° were used [for more information: https://www.fao.org/3/x0490e/x0490e0j.htm#annex%202.%20meteorological%20tables].

The root space competition was determined by the screen projection surfaces. For this purpose, the tree cadaster (ESRI shape) was first expanded by the crown diameter and then blended with the already prepared individual layers of the examined tree species. The calculated area required all overlapping areas of the crown projections areas of the TOI and its corresponding competitors. Subsequently, the areas for the respective tree of interest were summed. The R extension package for QGIS, specifically the script “summarize by two fields”, was used to summarize the single overlay areas for all competitors to the TOI ( ∑ A O , j ) using the “doBy,” “foreign,” and “plyr” packages. R K = 1 − A s t d ; T O I − ∑ A O , j A s t d ; T O I + ∑ A O , j 2 . (31)

Assuming a crown-root ratio of 1:1, the root expansion from TOI was approximated, based on the crown diameter using the following equation. A s t d ; T O I = π 4 ∗ c d 2 . (32)

The competitive pressure of the TOI and its corresponding competitors was considered according to Equation 33: K e = S D K + R K 2 . (33)

Parameterization by multiple regression analysis was performed as follows: ln A i = ln A n + α ∗ ln a g e i a g e n ∗ T e + β ∗ ln a g e i a g e n ∗ N e + γ ∗ ln a g e i a g e n ∗ K e . (34)

The individual parameters were related to general growth. For a g e i − a g e n = 0 , the restriction for the following equation is given by A i = A n ; a summary of the parameters is as follows: L U = α ∗ T e + β ∗ N e + γ ∗ K e . (35)

The extent of the allometric relationships for the tree of interest is estimated after transformation as follows: A i = A n ∗ e ln a g e i a g e n ∗ L U . (36)

In the previous equation, the parameters A n and a g e n assumed one.

To interpret the prognosis results, we used the mean absolute error MAE and the root mean square error RMSE. With the MAE, the outliers in the database are not weighted as heavily as they are with the RMSE. Thus, a comparison of the two measures of accuracy (MAE and RMSE) of the estimators gives a good overview of the most suitable function for the respective parameters. M A E = ∑ A n − A i n , (37) R M S E = ∑ A n − A i 2 n . (38)

The MAE and RMSE were evaluated using rank analysis. The lowest MAE/RMSE was assigned the lowest rank; accordingly, ranks 2 to 9 were assigned in ascending order for increasing MAE/RMSE. The estimator with the lowest rank was assumed to have the best forecasting performance. This, in turn, allowed a conclusion to be drawn regarding the abiotic factors (temperature and precipitation). For all investigated allometric correlations, rank analysis was performed and the mean value was calculated over the respective results.

The results of the regression showed a variety of relationships between allometric parameters and the abiotic environment. Trees in parks are generally smaller than those located on streets. Local climatic conditions could have caused this effect. The water uptake is often lower and the thermal conditions are often bitter, unlike trees in parks. This may be due to faster growth.

It was not possible to identify the expected correlations for all tree species. Consideration of the bhd in all nine scenarios for A. campestre showed that when approaching the limit value (0.6 in our example), the bhd decreased (negative coefficient). Since this coefficient was not significant in any scenario, a correlation between the temperature optimums chosen here must be discarded. However, it was evident that the water supply was important in many scenarios. A better water supply resulted in a larger bhd. Similarly, a better water supply meant less height growth. However, competition had a beneficial effect on height growth. Acer platanoides showed a significant correlation with bhd growth, especially in the scenarios with higher temperature optimum. Regarding height growth, competition resulted in a reduced height gain. Likewise, the effect of a lower height increased with a better water supply analogous to Acer campestre was also evident here. No significant correlations for an optimum temperature range between 17°C–19°C were observed. Malus sp. showed a significant correlation between bhd growth and temperature in all scenarios. A higher temperature encouraged greater growth. Precipitation was not significant in any scenario but showed a negative impact. The competition showed significant correlations with both bhd and height. Greater competition had an inhibiting effect on height growth. On the other hand, Q. robur showed significant correlations between temperature and bhd and altitude growth across all nine scenarios. Thus, a greater relative temperature optimum had a positive effect on growth. No significant correlations were observed for water supply.

To improve the forecast quality, further data are needed for verification. Furthermore, the growth function requires adjustment. Accordingly, the different phases of tree life can be used to adapt to the initial growth. Accordingly, better solutions can be achieved iteratively by extending the optimums or adjusting Equation 5. In addition, changing the weighting of the respective allometric relationship (thickness, height, and crown growth) can affect the results of the rank analysis. The crown growth of street trees is particularly influenced by regular pruning; thus, this growth is influenced by humans with recurring regularity.

Missing parameters that might further influence tree growth can be identified through future data collection. The implementation of sensor technology can be used to gain detailed knowledge of physiological responses to environmental changes and the potential associated effects on allometric contexts. A better prognosis for site-specific ecosystem services can be achieved when the growth characteristics are sufficiently known. In the future, by focusing on the provision of ecosystem services in municipal planning, the developed routines can be used for detailed sustainable urban greenery planning. The challenges to the application of sensor techniques to public green spaces might be theft and vandalism. The permanent installation of sensors in publicly accessible areas is also challenging.

The ecosystem performance of carbon sequestration is highlighted when trees act as urban carbon sinks. Oxygen is vital to most organisms. Photosynthesis is a crucial biochemical process that forms the basis for carbon sequestration. This process depends on the plant species, temperature, CO₂, water uptake, and especially light consumption. If the corresponding parameters are collected from many trees of a certain species, existing knowledge can be upgraded. In particular, inherited adaptations to the respective site conditions during natural spread are interesting. Water uptake and temperature are easily quantifiable parameters. At this point, the related ecosystem services can be measured along with physiologically relevant parameters.

Further insights can be gained by measuring the solar radiation in the foliage at different heights and in the surrounding area. These measurements can be used to assess summer mortality. Moreover, statements regarding comfort can be made in this setting. The sensors must be integrated representatively with reference to the leaf area to identify corresponding relationships.

Foliage and crowns are essential components in determining the dust filtering potential. Leaf placement, integral roughness, and the porosity of the leaf area should be considered in detail. These measurements should be performed with a durable integrated dust sensor in tree crowns and with reference to the immediate environment. In addition, a regular sensor should be implemented to collect leaves. This can be followed by dry and wet quantification methods.

Cultural values are notoriously difficult to quantify. However, their impact on human well-being is undisputed. The flowering of Prunus serrulata can illustrate the importance of green infrastructure for social components. Avenues with these trees serve as popular photo motifs (own observation; see Figure 24 Holzweg-Magdeburg).

However, urban trees might also cause disadvantages, allergies, and damage to gray infrastructure, which are nuisances for humans. The hazards resulting from trees are often announced in advance. A longer growing season and warmer conditions promote faster growth of individual trees and represent an additional stressor at the extrema. Particularly, the production of biogenic volatile organic compounds (BVOCs) is associated with stressful situations. If the highly complex language of trees can be deciphered, appropriate countermeasures would be more promising. In addition to chemical messengers in tree communication, inner cries provide truth signals. These noises are only perceptible in the ultrasonic range and are believed to be caused by the tearing of the water column (cavitation).

Overall, far-reaching findings have been obtained in recent years. However, many research questions remain to be addressed in the future, such as the impact of heavily anthropogenic soils on tree growth.