A Proposal of a Botanical Tree Growth Control Method by Hormone Distribution Supplied from the Root

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1 A Proposal of a Botanical Tree Growth Control Method by Hormone Distribution Supplied from the Root Ji-Joon Kim Nanzan University kim@it.nanzan-u.ac.jp Abstract In this paper I propose a growth control method for botanical tree models controlled by virtual hormone distribution that supplied from the root. Natural trees grow under the control of water, nutrients and plant hormone distribution, and some of growth models aiming to represent realistic tree growth by incorporating those features were proposed. To reproduce old trees that have a long life more than several hundred years, long term growth simulation is needed. However, there is the possibility of growth failures such as over branching and/or few branching by using existing models. On the other hand, it is possible to control for long term growth by fitting to predetermined parameters such as logistic curves. However, this method is difficult to represent recovery growth form prunings and/or injuries of a tree. I propose a growth control model of trees. In this model, a virtual plant hormone is supplied from root and distributed in propotion to the number of twigs. In addition to the acropetal hormone, another basipetal hormones are supplied from apices of new twigs and flow into the root. Then the branchings and the elongation of twigs are controlled autonomously by the distributed acropetal hormone, and old branches are removed by the basipetal hormone.the hormone behaviors aren t based on real plant hormones, but it is easy to control tree growth and it has ability to represent long term tree growth more than hundred generations. 1 Introduction CG simulation of natural objects and natural phenomena including botanical trees has become popular in recent years. A tree, a complex object, is necessary to represent natural scenes by CG, and is used in diverse fields such as computer games, movies, and visual simulations for garden environments. Several models of production for trees have been proposed. The early models generate branching patterns using fractal and/or rules of formula grammer [6, 1, 7]. Some recent models are based on the features of real tree growth and integrated the effects of the environment, nutrients, and/or plant hormones [3, 2, 5, 8, 4]. These models have been mainly used to represent relatively younger trees that have several decades of age, and may cause growth failures in long term growth such as over branching and/or few branching. For example, younger trees branch frequently and elongate rapidly in a short periond, while older trees have nearly fixed number of branches and many of twigs are often witherd and fall off. To represent these features, consideration about aging of trees is necessary. On the other hand, there is a method to represent long term growth by fitting to growth functions of living things such as logistic curves. However, the method enforces fitting to predefined growth curves, and it is difficult to represent recovery growth after a tree is pruned and/or injured. Some proper method to represent tree aging is expected that controling tree growth by under the consideration of plant hormones, nutrients, environment factors, etc. same as real trees. However, correct modeling of these factors is still difficult because of these complex correlations. I paid attention to tree growth itself. A tree is modeled as a layered structure of shoots such as buds, branches, and inflorescences. The position of branch apices are apart from the roots as progress of tree growth. I propose a growth control model of trees by taking advantage of this basic fact. In this model, a virtual hormone produced on the tree root is transmitted upward to the branch apices and the quantity is attenuated by the ratio of the transmission length and branching frequency. The number and the length of new twigs are controlled by the acropetal hormone quantity. In addition to the acropetal hormone, another basipetal hormone produced on the branch apex of new twig is transmitted downward to the tree root and the quantity is also attenuated as similar as the acropetal hormone. Old lateral branches are removed by the basipetal hormone quantity. Although these hormone behaviors and control methods aren t based on real plant hormones, it makes easy to control tree growth because of its simplification. I also verified that the tree height and the number of branches are controlled properly and it is possible to represent stable tree growth for the long period by the proposed model.

2 : Bud φ φ θ : main axis : lateral axis θ : divergence angle φ : branching angle : Shoot : Node Figure 3: Geometric Structure of a Binary Tree Figure 1: Tree Growth by Layered Structure of Shoots : main axis : lateral axis Figure 2: Tree Structure by Binary Tree of Nodes 2 Branching Model Generally, botanical trees are modeled as a layered structure of shoots (Figure 1). A shoot consists of several internodes and nodes. Leaves and buds that may change new shoots or inflorescences in next generation are formed on the nodes. The author previously proposed a tree growth model that produce tree shape by this layered structure of shoots, as a unit of a tree. In this research, a tree structure is modeled as a topological binary tree (Figure 2). The binary tree is a link structure of nodes. This reason is that the shoot of real plants express many kinds of variations such as the number of nodes, internode lengths, the number of buds etc., and many of parameters are necessary to represent those various kinds of shoots. A node may be divided into two new nodes. One new node will become the main axis, and another node are determined to be the lateral axis. A geometric structure of the binary tree is shown in Figure 3. A main axis grows in a direction parallel to the parent axis. Lateral axes diverge from the main axis at a branching angle φ and the growth direction is rotated around the main axis by a divergence angle θ. 3 Virtual Hormone As a method for tree growth control, I introduced two kinds of virtual hormones. A kind of virtual home is an acropetal hormone produced on the tree root and transmitted upward to the branch apices. Another one is a basipetal hormone produced on the branch apices and transmitted downward to the root. The acropetal hormone is used to control growth length and the number of new twigs of a tree, and the basipetal hormone is used to remove old branches. The acropetal hormone quantity U is attenuated by a fix ratio on each node. If a node has two child node, the hormone quantities of child nodes are attenuated again by different distribution rates either main axis or lateral axis. The hormone quantity produced on the root U root is controlled by the number of apices of twigs. The main axis grows stronger than lateral one by the different quantity of hormones. If U is lower than a threshold value, the node may create only one new node or none of new nodes. The node that can t create new nodes will be deleted. The basipetal hormone quantities D produced on the apices are same fix value and attenuated by a fix ratio on each node. The quantities are also attenuated again by different distribution rates as similar as acropetal hormones. If D is lower than a threshold value and the node is on the lateral axis, the node and the child nodes will be removed. 4 Growth Algorithm The growth algorithm of this model is described by the following steps. 1. Create initial stem with initial length L from the root node. 2. Set the maximum number of tip nodes N max to Determine the acropetal hormone quantity generated on the root node U root by the following equation: U root = U N max (1) where U initial acropetal hormone quantity, N max max. number of tip nodes.

3 4. Determine the basipetal hormone quantities generated on the each apex U apex by the following equation: D apex = D (2) where D initial basipetal hormone quantity. 5. Distribute acropetal hormone from root node to tip nodes recursively by following equation: U n = αβu n 1 αγu n 1 αu n 1 for main axis, for lateral axis, for single axis, no distribution. (3) where U n 1 acropetal hormone quantity of a parent node, U n acropetal hormone quantity of a child node, α attenuation rate for each node, β distribution rate for the main axis, γ distribution rate for the lateral axis. 6. Distribute basipetal hormone from to tip nodes to root node recursively by following equation: D n = where { λ(µd main n+1 + νd lat. λd n+1 n+1) two child nodes, one child node. (4) Dn+1 main basipetal hormone quantity of a child node on main axis, Dn+1 lat. basipetal hormone quantity of a child node on lateral axis, D n+1 basipetal hormone quantity of a child node, D n basipetal hormone quantity of a parent node, λ attenuation rate for each node, µ distribution rate for the main axis, ν distribution rate for the lateral axis. 7. Remove old nodes having basipetal hormone that quantity is lower than threshold value D remove recursively. 8. Determine the number of each branching node N new from tip node by following equation: U n < ɛ main N new = 1 ɛ main U n < ɛ lat. (5) 2 U n > ɛ lat. Acropetal Hormone Quantity α=.8 α=.85 α=.9 α=.95 Figure 4: Attenuation of Hormone Quantity on the Tip Node where ɛ main branch threshold for main axis, ɛ lat. branch threshold for lateral axis. 9. Determine the lengths for new nodes by the following equation: L new = U n L (6) 1. Create new nodes determined by equation Redetermine N max by following equation: N max = max(n max, N n ) (7) where N n is the number of new tip nodes. 12. Remove old nodes that have no child nodes recursively. 13. Go to step 3. 5 Simulation Results The simulation results of the proposed model are described in this section. The tree structure of the results illustrated in figures 4 and 5 is constructed a series of nodes, not divided, and basipetal hormone is not used. Figure 4 shows the attenuation of acropetal hormone quantity on the tip node. Figure 5 indicates the fluctuation of tree height. The acropetal hormone quantity of the tip node decreases with increasing generation number. The increasing ratio of tree height is also decreases. Therefore, the maximum tree height will be convergenced to the fixed length. The tree structure of the results illustrated in figures 6 and 7 is constructed as a binary tree. Figure 6 shows

4 Maximun Tree Height α=.8 α=.85 α=.9 α=.95 Acropetal Hormone Quantity Max. Acropetal Hormone Min. Acropetal Hormone The Number of Nodes Figure 5: Limitation of Tree Height Without Control Controlled by U Controlled by D U and D Figure 7: Transition of the Max. and Min. Acropetal Hormone Quantity with Acropetal Hormone Control Acropetal Hormone Quantity Max. Acropetal Hormone Min. Acropetal Hormone Figure 6: Transition of the number of nodes the transition of the number of nodes. The number of nodes without control is proportional to the square of the generation number. The increase of the number of nodes will be inhibited by hormone control. The number of nodes controlled by acropetal hormone U is increasing quadratically in young period, then it is gradually decreasing with vibration. The number of nodes controlled by basipetal hormone D is increasing as similar as no-controlled result in young period, then the number is fixed. Under the control of two hormones, the number of nodes is lower than previous results because of removal of old branches. However, the number is maintained substantially constant. Figure 7 and 8 indicates the transition of the maximum and minimum acropetal hormone quantities. The result of figure 7 is controlled by acropetal hormone, and the result of figure 8 is controlled by two hormones. In young period of figure 7, main axis receives much of acropetal hormone quantity and grows rapidly. After that, the quantity is controlled to the fixed value with some vibration. Meanwhile, the minimum acropetal hormone quantity supplied to lateral axes decreases rapidly Figure 8: Transition of the Max. and Min. Acropetal Hormone Quantity with Two Hormones Control in young period, then the quantity is also controlled with a little vibration. The minimum acropetal hormone quantity in figure 8 shows as like as figure 7. On the other hand, the maximum acropetal hormone quantity in figure 8 is larger than figure 7 and controlled with large vibration because of the removal of old branches. The tree images of the results illustrated in figure 9, 1, and 11. Figure 9 shows the result by controlling acropetal hormone. Figure 1 illustrates the result by controlling basipetal hormone. And figure 11 shows the result by controlling two hormones. In figure 9, the number of tip nodes is controlled but there are a lot of old branches. Meanwhile, old branches in figure 1 are removed but the young part of a tree structure is as similar as binary tree. In figure 11, both the number of tip nodes and old branches are controlled simultaneously. However, the overview looks like as figure 9 because of removal of old latreal branches. The set of parameters for the simulation results are

5 Figure 9: Tree images with acropetal hormone Table 1: Parameters for the Simulations Symbol L U D α β γ λ µ ν Dremove ²main ²lat. θ φ No Branching Branching listed in Table 1. 6 Concluding Remarks I proposed a control method for the long term growth simulation of botanical trees in this paper. This model controls branching and elongation of a tree by the quantity of virtual acropetal hormone supplied from the tree root and transmitted upward, and controls removal of old branches by the quantity of virtual basipetal hormone supplied from the apices of new twigs and transmitted downward. As the results of growth simulation of trees, I also verified that the model has ability to keep branch number and to control tree growth for more than handred generations. However, there are some improvemnets in this model. Firstly, a structure of a tree produced by this model only shows as similar as a conifeous tree. I am planning to improve hormone control methods to produce a structure of a broad leaf tree. Secondly, this model doesn t considering the shoot structures of real trees. I intend to conbine this proposed model and the former model that can represent Figure 1: Tree images with basipetal hormone shoot structure[4], and check the ability to represent real old tree structures. Thirdly, this model uses two kinds of hormones. However, real tree growth is affected by various hormones. In a real tree, some basipetal hormones are produced on the tip of twigs and acropetal hormones are produced on the root tips in underground. Therefore, I am planning to improve hormone models and botanical root system in underground to represent growth control of whole tree shapes. Furthermore, it is important to integrate environmental models such as light, nutrient, environment factors. Acknowledgments This research is supported in part by a Pache Research Subsidy (No. I-A-2) from Nanzan University. References [1] M. Aono and T. L. Kunii. Botanical tree image generation. IEEE Computer Graphics & Application, 4(5):1 34, May [2] N. Chiba, S. Ohkawa, K. Muraoka, and M. Miura. A growth model of botanical trees generation of natural hsapes of trees based on an imaginary plant hor-

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