## Methods
### Classification-Based Tree Matching and Co-Registration
#### 3.6.3 Classification-Based Tree Matching  
Following the idea of feature descriptors, the proposed tree matching method derives features which have the potential to classify each potential tree pair \((s,d)\in(S\times D)\) of surveyed trees \(S\) and detected trees \(D\) as either _Matching_ or _Not matching_. These features serve as input for a Random Forest [42; 43] classifier to estimate the probability \(p_{s,d}\) of the class _Matching_ (hereafter \"matching probability\"). Caused by GNSS errors, a shift between the surveyed trees \(S\) and detected trees \(D\) has to be assumed. Thus, the displacement vector between both point sets \(S\) and \(D\) has to be estimated. Assuming a surveyed tree \(s\in S\) corresponds to a detected tree \(d\in D\), the displacement vector is approximately \(d-s\), and we can perform a GNSS correction by applying \(T_{d-s}S\). Since we can assume displacements similar to the GNSS errors, the probability \(p_{s,d}\) should decrease with increasing values for \(|d_{x}-s_{x}|\) and \(|d_{y}-s_{y}|\). Further, we assume the surveyed tree \(s\) matches the detected tree \(d\), which results in almost similar point patters \(T_{d-s}S\) and \(D\). Thus, we can find a set of \(k\) neighbored tree pairs \(r^{\omega^{F}(T_{d-s}S,D)}_{\leq k}\) which is characterized by point pair distances \(\|r^{\omega^{F}(T_{d-s}S,D)}\|_{2}\) (Equation (10)) close to zero. Note that the weights \(\omega^{F}(T_{d-s}S,D)\) (Equation (9)) result in a tree pair assignment in ascending order of distance. Missing tree pairs (which occur if \(k>min(|S|,|D|)\)) are indicated with infinite weights. Assuming \(s\) does not match \(d\), then we would expect larger point pair distances, because here the point patterns will most likely differ. Thus, the point pair distances \(\|r^{\omega^{F}(T_{d-s}S,D)}\|_{2}\) serve as a sequence of features to indicate similar point patterns. \[\omega^{F}(A,B)=\left(\omega^{F}_{a,b}= \begin{cases}\frac{1}{\|b-a\|_{2}},& \text{if }a\
eq b\\ \infty,&\text{otherwise}\end{cases} \right)_{a\in A;\ b\in B} \tag{9}\]  
\[\|r^{\omega(A,B)}\|_{2}=\left(\|r^{\omega(A,B)}_{i}\|_{2}\right)_{i\in\{1, \ldots,|r^{\omega(A,B)}|\}} \tag{10}\]  
For a given sequence of matching tree pairs \(r\), an almost one-to-one correlation between the tree heights \(h_{S}(r)\) and \(h_{D}(r)\) (Equation (11)) should be observable. Mismatched trees should lead to uncorrelated tree heights. Thus, the Pearson's correlation coefficient \(\rho\) of these heights serves as an additional feature of the pair \((s,d)\). \[h_{S}(r)= (s_{h})_{(s,d)\in r} \tag{11}\] \[h_{D}(r)= (d_{h})_{(s,d)\in r}\]  
In addition to these indicators for similar point patterns, additional features are derived which give information about uncertainties associated with a potential tree pair. Since the probability of identifying a correct tree pair by chance decreases with an increasing number of potential tree pair combinations, the number of trees \(|S|\) and \(|D|\) serve as additional features. Since close-standing trees complicate the unambiguous identification of matching tree pairs, the average nearest-neighbor distances \(\overline{NND_{S}}\) and \(\overline{NND_{D}}\) (Equation (3)) are calculated as an indicator of tree separability. Based on the given features, a feature vector \(f_{s,d}\) (Equation (12)) is calculated for each potential tree pair \((s,d)\in(S\times D)\). Figure 4 visualizes the workflow for deriving such a feature vector. It also illustrates how the set \(S\) is relocated by applying the translation matrix \(T_{d-s}\) to assign the neighboring tree pairs \(R^{\omega^{F}}(T_{d-s}S,D)\). \[f_{s,d}=\left( \begin{array}{c}\|d-s\|_{2}\\ d_{x}-s_{x}\\ d_{y}-s_{y}\\ d_{h}-s_{h}\\ \|r_{2}^{\omega^{F}(T_{d-s},S,D)}\|_{2}\\ \vdots\\ \|r_{k}^{\omega^{F}(T_{d-s},S,D)}\|_{2}\\ |S|\\ \frac{|D|}{NND_{S}}\\ \rho_{h_{S}\left(r_{z\leq k}^{\omega^{F}(T_{d-s}S,D)}\right),\,h_{D}\left(r_{ z\leq k}^{\omega^{F}(T_{d-s}S,D)}\right)\right)}\end{array} \right.