# Constrained Multi-Objective Optimization of Simulated Tree Pruning with Heterogeneous Criteria
## 2 Materials and Methods
### EduAPLE Tree Growth Simulator  
Tree growth simulator EduAPLE [7; 15] was developed as a software tool that allows the user to observe the effects of various tree training techniques, such as pruning, weighing, and spreading, on the formation of an apple tree crown. To this end, EduAPLE implements a parameterized development model that allows simulation of different growth behaviors of apple trees. EduAPLE uses a simplified tree structure representation, where the tree skeleton is composed of short linear segments called metamers, which are linked to form branches. Metamers also present the basic unit of increment in growth simulation. Eachmetamer consists of an internode and two buds, a terminal and a lateral one, which are used to extend or fork new branches upon growth. Internodes have assigned unique identifiers, which are used to indicate the cut locations for pruning. Figure 1 shows an example of a simple tree structure with depth-order internode labeling.  
Simulation of tree growth in EduAPPLE is based on a source-sink model, and performed in discrete seasonal steps. The yearly increments are determined by first calculating the growth resources accumulated by a tree, and then redistributing them to different parts of the tree. The growth is realized by replacing the shooting buds with new metamers. The main sources of growth material are the tree's own food reserves and photosynthesis. The amount of food reserve is modeled in EduAPPLE as a linear dependency on tree age \(A\), while the quantity of photosynthetic product is estimated from total bud illuminance \(q_{i}\):  
\[R=C_{1}\min\{A,12\}+\left(C_{2}\tanh\left(\frac{0.2}{A}\right)+2\right)\sum_{ i\in\mathcal{B}}q_{i}. \tag{1}\]  
Here, \(\mathcal{B}\) denotes the set of all buds, while \(C_{1}\) and \(C_{2}\) are adjustable model parameters. The hyperbolic tangent non-linearity is used to limit the tree growth with age. Bud illuminance is computed by estimating the amount of shadow the bud receives from other tree elements. The details of this calculation can be found in the original EduAPPLE paper [15].  
Following the allocation theory [28], the accumulated resources \(R\) are split into two exclusive portions, which are dedicated separately to vegetative and reproductive growth. The estimated reproductive requirements of a tree are calculated using the number \(N_{f}\) of flower buds as [13]:  
\[r_{\text{f}}=N_{\text{f}}(C_{3}-C_{4}\times A). \tag{2}\]  
Here, \(C_{3}\) and \(C_{4}\) are the allocation model parameters that can be adjusted to obtain specific behavior. The rest of the resources are allocated to vegetative growth. The reproductive pressure on resources can be reduced by pruning, which controls the balance of reproductive and vegetative growth across seasons.  
Once determined, the vegetative growth resources are redistributed back to the buds in proportion to their light exposure, orientation, and distance from the roots. The probability of shooting is higher for well lit and vertically oriented buds, but reduced for older buds. Given the final amount \(r_{i}\) of growth resources, bud \(i\) can sprout a sequence of \(|r_{i}|\) new metamers. Upon creation, new buds are differentiated into vegetative and flowering ones, where the probability \(p_{f}\) of a bud becoming a flowering one is another model parameter.