3.2. External Factors Influencing the Growth of Undergrowth
In each plot, we estimated the composition of the first storey, the density of the first storey, the covering of underbrush and herbaceous plant, and forest site. These indicators were assessed as external factors that influence the undergrowth of oaks. We found that the average density of the first storey in Trakas Forest was 0.6, the covering of underbrush is 16.1% and the covering of herbaceous plants is 21%. In the first storey, we found 11 species of trees. The biggest part of stand was composed of oaks (pedunculate, sessile and their hybrids), 20.7%; Norway spruces, 32.7%; Scots pines, 24.3%; and silver beeches, 10.2%. More than one of 4 tree species were present: aspen, 3.9%; common hornbeam, 2.3%; small-lived lime, 2%. Norway maples accounted for 0.5%, and European ash and European larch did not amount to 0.1%. After analysing the influence of the forest site on undergrowth abundance, we determined that the greatest number of oaks in the undergrowth were in mesoeutrophic soils of normal moisture (Nm) and eutrophic soils of normal moisture (Ne) (
Figure 4). The quantity of sessile oak was greatest in mesoeutrophic soils of normal moisture on slopes (>15°) soils (Sm) site, and the quantities of pedunculated and hybrid oaks were greatest in Nm sites.
The abundance of oak undergrowth was largest where spruces, pines, and beeches were predominant in the first storey. There was a bit less oak undergrowth in oak sites, and there was very little or none where aspens, alders, limes, and hornbeams were growing in the first storey (
Figure 5). The greatest quantity of oak undergrowth in the first group was found where spruces and pines predominated in the first storey. The greatest quantity of the second oak undergrowth group was in the beech and pine sites, the third one in beech and oak sites, and the fourth one in beech and pine sites. The abundance of oak undergrowth is distinguished by individual species of oak. For example, the undergrowth of pedunculated and hybrid oaks was more in the site where beeches, pines and spruces, and sessile oak—where oaks (in the 4 group—pines) dominated in the first storey.
Redundancy analysis (RDA) estimated the coefficients of oak undergrowth abundance depending on density of the first storey, covering of underbrush and herbaceous plants (
Figure 6). The RDA showed no linear dependence between x and y variables among all the oak undergrowth, but it was very small linear dependence among hybrid and sessile oaks undergrowth.
Hybrid origin oak undergrowth axis distribution is more similar to that of sessile oak. The density of the first storey had a negative impact for the second undergrowth group’s recovery. Higher underbrush and herbaceous plant covering impact reveal only sessile oak undergrowth. The data showed that fitocenoid structure of the elements influenced oaks of the different height groups unequally. In this case, it would better to analyse by different height groups of undergrowth. Kremer et al. [
28] researched results of the process of sessile oak recolonization when one species spreads and becomes established in a certain territory. They found that hybridisation generates other species during generation changes. It is a process that lasts for centuries, in the course of which the oak grows up and starts to prevail in the stands where only a pedunculate oak grew. This leads to increased pollination from pedunculate oak isolation. In summary, the proportion of sessile oaks in the species composition of stands will increase in the future as new sessile oaks are detected in the undergrowth where the first storey of the forest does not contain mature oaks. This conclusion is further confirmed by the spreading of hybrid origin oaks in Trakas Forest, keeping in mind that hybrids are more easily hybridized with sessile oak than with pedunculate oaks.
3.3. Mathematical Modelling of Oak Undergrowth
The number of all founding oaks (
Np,g) are summarizes in the mathematical modelling data set. The total results of undergrowth quantity included 4 × 4 size distribution, where interdependence of elements can be evaluated with
criterion [
30]. Such volume statistics, with 6 degrees of freedom at the 0.05 significance level, were
. The quantity of all total undergrowth oak species and age groups was interdependent because statistic
is greater than critical. The total undergrowth quantity was an expression of the extensive system properties that can be characteristic only of the study system.
The structural analysis of undergrowth in Trakas Forest showed that a half of all oaks were included in the first height group (up to 0.5 m height). Their quantity was more than 2 times greater than the quantity of the 2nd and 4th height groups of oak (0.5–1.5 m and upper 3 m height). An exception was the 1.5–3 m height group, in which the quantity of oaks did not reach 15% of the quantity of smallest oak height group.
The spreading of undergrowth in the forest characterizes the number of temporary plots where only one oak was found. There were 141 such plots, about 70% of the total number of plots. This suggests that process of oaks' natural regeneration goes evenly. Natural forest covering was even when target tree species account for more than 66% of all plots [
31].
The territorial distribution of undergrowth species was rather uneven. The number of plots with sessile oak undergrowth was only 9%. Meanwhile, hybrid oak undergrowth was in 37% of research plots. Statistical analysis showed that the number of plots with oak undergrowth was not interconnected, because statistic was smaller than critical meaning.
The density of undergrowth by systematic point of view was intense, which characterizes its quality. It could not be applied the additivity rule. Therefore, the distribution of undergrowth density was characterized by mean value calculated according to Formula (2). The observed maximum undergrowth density was several times higher than the estimated average of its value (
Table 1). They were of the same order of magnitude and fully compatible with undergrowth density values observed by other authors. We determined that in places where oaks regeneration is on-going, the middle number of oaks is 4389 un./ha. This number of undergrowth self-regeneration oak seedlings was closer to the number of oaks in violet-wood sorrel (
hepatico-oksalidosa) forest type undergrowth of pure oak forests—5200 un./ha [
6]. The average density of the data was more generally characteristic and less dependent on space surveillance and better describes the total Trakas Forest oak undergrowth condition, although it did not reveal the reasons such a condition has occurred. Undergrowth of medium density results, grouped according to their formation time, can be used for different kinds of undergrowth spreading characteristics of the evaluation.
In oak undergrowth formation process, the data, grouped by formation period, revealed that oak undergrowth changes over time. Regrouping the total amount of undergrowth showed a growth compared to the amount of undergrowth in the earliest formative period. Observation time (
) found the total amount of oak undergrowth from the beginning of its formation had increased about 5 times (
Table 2). Growth of sessile and hybrid undergrowth number of oaks decreased more slowly.
In a similar way, the plots of oak undergrowth characterized the spread of forest undergrowth area development (
Table 3). The undergrowth area investigated during the life of maximum height oaks expanded about 1.75 times. The data showed that the sessile oak undergrowth spread in the territory progressed by 5% and hybrids even faster, by 11%, than the pedunculate oak. A pedunculate and hybrid oak undergrowth territorially formed in the same places, and their common area increased by only 1.4 times.
The change of quantity of undergrowth and covered territory over time was determined by the investigated undergrowth of four height groups and the qualities described by undergrowth density, calculated according to Formula (7) (
Table 4). The calculation shows that Trakas Forest undergrowth’s qualitative characteristics identify the pedunculate oak undergrowth as the dominant mature tree species in the stand. At the time of the study, there were 1047 trees per hectare. The presence of hybrid oak increases this indicator by 3% and, with the addition of sessile oak, by 6%. The density of hybrid and sessile oaks undergrowth there were about 2.5 times less than pedunculate oak.
A more detailed picture of the evolution of the undergrowth formation can be obtained only by mathematical modelling. The main purpose of such mathematical modelling was to define the main factors that determine this process. Both regression analysis to function
f (
Ap,t) in Equation (7) and analysed variables chosen should be compatible with the characteristics of the test process, and the validity of the obtained dependencies can be solved only on the basis of Fisher criterion (
F) and coefficient of determination (R
2) values [
30]. Therefore, regression analysis was performed on pedunculate and sessile oak species and their hybrids and on the whole oak undergrowth depending on the period and the formation of all possible interactions between variables (
Table 5). In the regression analysis, for the purpose of better images, non-absolute values of the density of the oak densities of
Ap presented in
Table 5 were used, and their relative values at the time of observation (
t = 0) were captured in terms of values.
First, let us look at all kinds of oak undergrowth relative density
function of time
t factor regression analysis of the results presented in
Table 5. Total undergrowth evolution can be described not only as the exponential:
but also as the Gompers type of dependence (R
2 = 0.996):
The graphic image of such dependence on the time factor (
Figure 7, curves 1 and 2) showed a good coincidence with the monitoring data and basically does not have any major advantages than the more complicated Gompers model. Attempts to analyse total oak undergrowth density dependence of two variables basically only confirmed the interdependence (
F >
Fkr) undergrowth of hybrids (
Table 5, line 13). Similar interdependences were determined for undergrowth of pedunculate oak (lines 2 and 3).
For the undergrowth spread, peculiarities analysis was limited to the provided opportunities of monitoring data group exponential regression analysis by depending only on its formation period.
Table 5 shows the undergrowth relative density regression analysis, revealing that only pedunculate oak understorey was statistically significant (
F >
Fkr), although the data determination coefficient was high in all studied oak trees:
The theoretical average undergrowth density dependences (10)–(12) were substantially different only in the exponent indicator in the size parameter m1 by relative approach. The largest average density increase rate was for the undergrowth of pedunculate oak and smallest for sessile oak. The total oak undergrowth density increase rate was higher than the values obtained by individual components (Equation (8)). That strengthening of oak undergrowth density increase rate as one formation property was significant, indicating the effect of the biological system.
The graph image of this research (
Figure 7) only additionally confirms the differences of undergrowth average density increase rates of individual undergrowth species and the total oak undergrowth. The higher distribution of oak undergrowth density could be linked with effects of other factors not considered here (cuttings, diseases, etc.). We can neither deny nor confirm that the observations data could be affected by the cuttings in Trakas Forest a decade ago [
26]. Despite the fact that undergrowth formation peculiarities were studied only in the formation period, the duration of which was difficult to define, we can conclude that sessile oaks positively influence forest development, because the whole of the undergrowth was formed with higher rates than the individual components of the forest.
The numerical investigation determined that the average density increase of oak undergrowth elements was a significant effect of the system that formed in the course of natural interaction. This oak undergrowth processing feature determines the investigated stand tree species change in the long term.
Other authors’ pedunculate oak undergrowth studies had been limited with extensive features (e.g., the amount of undergrowth and so on), using other monitoring techniques, allowing comparisons to be made only at the level of change trends. In our study, an exponential regression analysis was carried out in
Table 4 to present undergrowth’s combined total quantities of all Trakas Forest undergrowth components. The analysis showed that the resulting exponential relative volumes of undergrowth were rather similar (
Table 6).
Why the sessile oak exponential rate was much less is related with the already discussed reasons. It can be concluded that in contrast to the average density rate, the relative data of oak undergrowth in respect of formation period varied equally. The pedunculate oak understorey received dependence
After evaluating the observed age (
a) of the oaks, depending on the formation period (
t) according to Equation (5), we get the dependence:
The graphic representation of this addition to the condition
was fully comparable with the data of other authors [
6], which were dependent on the type of oak forest (
Figure 8). It might be noted that the formative period of undergrowth could be much shorter than 5 years (
Figure 8, curve 1), but from such comparisons, it would be difficult to identify the latest undergrowth period. The age undergrowth was characterized by large variations depending on the type of oak forest, apart from the fact that the observations of other authors did not include all the forest areas. Here the observations on the undergrowth of the relative amount of dependence on the relative amount of undergrowth just shows that this indicator (character) as the extensive feature is more dependent on the species of the whole stand and does not change the relative parameter findings on the undergrowth spread rates. The functional expediency of ongoing processes in undergrowth should be seen in terms in applied forestry, taking into account the characteristics of hybrid oaks, resistance to diseases and pests, and so on.