PATH LENGTH AND HEIGHT IN ASYMMETRIC BINARY BRANCHING TREES

Transcription

1 PATH LENGTH AND HEIGHT IN ASYMMETRIC BINARY BRANCHING TREES AMELIA O HANLON, PATRICIA HOWARD, DAVID A. BROWN Abstract. This work is inspired by a paper by Mandelbrot and Frame [MF], in which they describe properties of symmetric binary branching trees. We study a variationof their trees in which non-uniform scalingis applied, focusing on geometric properties such as path length and tree height. We also discuss the computer implementation via Mathematica. Contents. Introduction 2. Path Length 3 3. Tree Height 7 4. Computer Implementation 0 5. Conjectures References. Introduction We study the geometry of asymmetric binary branching trees, a variation on the symmetric trees found in [MF], focusing on path length and tree height. Instead of using a uniform scaling ratio in the branching process, we chose to apply di erent scalings based upon whether branching occurred to the left or the right. These trees will be refered to as asymmetric branching binary trees, or simply asymmetric trees. This slight change leads to wealth of new geometric characteristics. In order to better understand asymmetric trees it is first necessary to consider their symmetric counterpart. To construct a symmetric branching tree, first construct a vertical segment which will be referred to as the trunk. For simplicity let the trunk have unit length. Then extend two branches from the tip of the trunk, each branching by an angle > 0 from the vertical extension of the trunk and having length r, where r 2 (0, ) is a scaling ratio. Two branches are then created from the tips of each of these branches in a similar fashion. At this stage, the four new branches will have Date: December 4, 200.

2 2 A. O HANLON, P. HOWARD, AND D. BROWN length r 2. This process is continued indefinitely [see Figure A]. Notice that the length of branches created at stage n is r times the length of the branches at stage n. Asymmetric fractal trees will be constructed in a similar manner except there will be two scaling ratios [see Figure ]. For simplicity will be the scaling ratio applied to right branches and r 2 will be the scaling ratio applied to left branches. Also, assume > r 2. The angle at which each branch is rotated from the extension of the previous branch (or trunk) will remain constant throughout the tree. By altering a symmetric tree in this way the tree no longer has left-right symmetry across the extension of the trunk but has self-similarity, meaning each small part of the tree is an exact copy of the whole tree. Figure. Comparison ofsymmetric andasymmetric branching trees. Both have = 0. A has r =, and B has r 2 =.9, r 2 =.2. In a fractal tree every branch can be described using a finite number of left turns and right turns. For example, the branch at the address T LR is found by travelling along the trunk, taking a left turn at the first branching and then taking a right turn at the second branching. Similarly a path within the tree can be described by the left turns and right turns contained in it. For example an infinite path that alternates right and left turns has the address T RLRL = T (RL) and a finite path contained in that path is T RLRL. In Section 2, we analyze the lengths of infinite paths emanating from the base of the trunk of the tree. The lengths of infinite paths can be determined using geometric series. In symmetric trees this series is + r + r 2 =. It is interesting to note r that every infinite path beginning at the base of the trunk in a symmetric tree will

3 GEOMETRY OF ASYMMETRIC BINARY BRANCHING TREES 3 have the same length. This results from the fact that all the branches constructed at the same stage have the same length. In contrast, the study of path lengths in the asymmetric branching trees is far from trivial due to the non-uniform scaling. Clearly, T R is the longest path, having length T R =, while T L is the shortest path, with length. In Section 3 we provide a complete analysis of tree height for asymmetric trees and give an algorithm for calculating this height. If the tree is placed on the Cartesian plane with the base at the origin, then the height is defined as the largest y coordinate in the tree. Although each path in a symmetric tree has the same length, they do not all reach the same height. Consider the path T R in Figure A. It is obvious that this path does not reach the greatest height in the tree. Now, consider branches T RR and T RL. Notice that it is the vertical branch that reaches the greater height. Therefore, by constructing a path with as many vertical branches as possible, e.g. T (RL) or T (LR), the greatest height can be obtained. So the total height of the tree, i.e. the height of the path T (RL) or T (LR), can be found using the equation: + r cos( ) + r 2 + r 3 cos( ) + r 4 + = + r cos( ). The paths T (RL) and T (LR) are also significant because they are examples of what will be referred to as opposite paths [see Figure 2]. In order to determine the address for a path s opposite switch the positioning of Rs and Ls in the address. For example, a path whose address begins with T RRLRL would have an opposite path whose address begins with T LLRLR. Also, T (RL) is an example of a periodic path as well, meaning the address consists of T followed by the pattern RL repeated infinitely. Another term that will be important later is corresponding branches. Consider that the branches T R and T L can be thought of as trunks of scaled down trees. Branches in the same positions in these trees are corresponding branches, as shown with branches f and g in Figure 2. Similarly, f corresponds to both branches h and i in di erent scaled down trees. This will be more thoroughly discussed in the next section. In Section 4, we discuss the use of Mathematica in drawing and analyzing the asymmetric trees. We give the basic outline of the use of the package created. Finally, in Section 5, we provide some conjectures and questions for further study. One of these topics is the study of space-filling curves, which arose in our study of trees. 2. Path Length Given an asymmetric tree, we can study the length, P, of a path P. Using geometric series, equations for infinite periodic path lengths are:

4 4 A. O HANLON, P. HOWARD, AND D. BROWN Figure 2. opposite paths T (R m L n ) = r m+ + m r 2 r n 2 m r n 2, T (L m R n ) = r m+ 2 + r 2 m r n m r n where 0 < <, 0 < r 2 <, and m, n 2 N [ {0}, not both 0. It is important to note that the second equation can be derived from the first by simply switching and r 2. This is because these paths are opposite [see Figure 2]. Other paths exist that are opposite, but whose lengths cannot be defined with the above equations (e.g. T RL and T LR ). One of the topics that will come up again and again is symmetry. Although these trees have an asymmetric branching pattern, they still contain self-similarity. Every branch can be thought of as the trunk of a scaled down tree. This lends to many interesting characteristics in an asymmetric tree. With the proper scaling ratio (combinations of and r 2 ) we can find the length of one branch using the length of another branch. For example, with a scaling ratio r 2, the length of any branch on the left side of a tree can be found using its corresponding branch length on the right side of the tree. From Figure 2 the length of branch f = r 3 is r 3 r 2 = r 2.,

5 GEOMETRY OF ASYMMETRIC BINARY BRANCHING TREES 5 Thus, the length of g (a corresponding branch of f) is 2 r 2. The branch T R 3 (f) corresponds to more than one branch in other trees within the right side of the main tree. Thus, f also corresponds to h(t RLR) and i(t RRL). Branches h and i are also of length 2 r 2. Self similarity in the asymmetric tree also permits us to study path lengths. By setting the formulas for path lengths of T RL and T LR equal to each other we can find a relationship between and r 2 in which these opposite paths are equal in length: T RL = T LR =) + = + r 2 =) = r 2 =) r 2 ( ) = ( ) Solving for, we find that =) r 2 + (r 2 r 2 2) = 0. = or = r 2. Thus, T RL and T LR are equal if + r 2 = or = r 2. The latter is the case of a symmetric branching tree, which we do not consider. Self-similarity in fractal trees implies that if a characteristic can be defined for the whole tree, that same characteristic can be found in each smaller tree within it. So, when + r 2 = there are infinitely many pairs of opposite paths equal in length stemming from their respective trunks. If a pair of opposite paths not stemming from the primary trunk are equal, those paths overlaying them that do stem from the primary trunk are equal in length, too. Each pair in this infinite set of equal path lengths corresponds to the pair (T RL, T LR ) and every other pair in the set. Three corresponding pairs are highlighted in Figure 4. Other sets of pairs of equal opposite paths (with a di erent length from the set previously discussed) can be found by setting the formulas for T R n L and T L n R equal to each other. These paths are equal in length if = ( r n 2 ) n or = r 2. Using generalized formulas for T (R m L n ) and T (L m R n ), we found a significant restriction on whether a pair of periodic opposite paths can be equal. Theorem 2.. If m n, where m, n 2 Z and m, n 0, then T (R m L n ) > T (L m R n ). Proof. For m, n 0, we find that

6 6 A. O HANLON, P. HOWARD, AND D. BROWN Figure 3 and T (R m L n ) = r m+ + rm ( r n 2 ) r m r n 2 T (L m R n ) = r m+ 2 + rm 2 ( r n ) r m 2 r n Now, since 0 <, r 2 < and > r 2, it follows that r p > r p 2 and rr p q 2 all p, q 2 Z with p q. Thus,. r p 2r q for (+ + +r m )+(rm +r m r rm rn 2 ) > (+r 2 + +r m 2 )+(rm 2 +r m 2 r2 + +rm 2 rn ), which implies that (+ + +r m )+(rm )(+r 2 +r rn 2 ) > (+r 2 + +r2 m )+(rm 2 )(+ +r rn ), and hence, r m+ + r m ( r n 2 ) > Since r mrn 2 < rm 2 rn, we find r m+ r m + ( r n 2 ) r m r2 n Therefore, T (R m L n ) > T (L m R n ). > r m+ 2 + r m+ 2 + r m 2 ( r n r m 2 ( r n ) r m 2 r n. ).

7 GEOMETRY OF ASYMMETRIC BINARY BRANCHING TREES 7 In continuing our studies of paths of equal length we had to resort to Mathematica. Often an enormous number of calculations had to be made in order to find values for and r 2 in which a pair of opposite paths were equal. This evaluation became too cumbersome. Studying path lengths that can not be defined with our equations for T (R m L n ) and T (L m R n ) may require a di erent approach. 3. Tree Height Tree height is defined by the height of the path that has the most vertical reach. When > 90 the trunk can be taller than its branches. This could possibly change the analysis of tree height as discussed here.(???) < 0 is essentially a reflection across an extension of the trunk. We studied tree height limiting our examination to trees in which 0 < apple 90. Thus, the question is: which path do we use to determine tree height? Weknow that the first right branch always has a greater vertical reach than the first left branch. Likewise, the right side of the tree is taller than the left side in asymmetric binary branching trees. This characteristic quickly becomes evident after only a few iterations since > r 2. The path can be determined by comparing the height of the path T R n to that of T R n L for n 2 (for n =, height(t R) will always be greater than height(t L)). This method of determining which path defines tree height is necessary since the height(t R n L) is not always greater than height(t R n ) (note: height(path) means the y value of the height of path)[see Figure 5]. Figure 4. Part of an asymmetric tree with = 3 4, r 2 = 4, = 0

8 8 A. O HANLON, P. HOWARD, AND D. BROWN The four quantities that determine whether a right branch extends above the left branch stemming from the same previous branch (or trunk) are, n,, and r 2. If = 90, then two of the paths determining tree height are T (RL) and T (LR) as they do in the symmetric tree. For every other between zero and ninety degrees the tip of a right branch can have a greater height than the tip of the left branch stemming from the same previous branch. Whether this occurs or not depends on the values for n and the ratio between r 2. In examining these three scenarios a pattern has developed. We can now find an inequality in terms of n that can be used to determine if the n th branch exhibiting greater height is a left or right branch: (3.) (3.2) (3.3) (3.4) height(t R n L) height(t R n ) () n cos((n 2) ) n cos(n ) () r 2 cos((n 2) ) cos(n ) () r 2 cos n cos(n 2) ). Not that equations [4.3] and [4.4] both prove to be e ective in determining which branch reaches greatest height in the tree. Thus, when the inequality is true, the end of the left branch attached to the (n ) th right branch has a greater, or equal, height than the n th right branch. If the left and right branches at the (n ) th branching are equal in height, then either path can be chosen in determining tree height. If they are equal and we chose the right branch, the next turn would be left. We know this will always be the case since two consecutive branchings will never have similar triangles when comparing their heights. This characteristic can be seen in the symmetric tree, as well. Once it has been determined that turning left on a path to greatest tree height is preferred to turning right, then the path of greatest height can be determined without further examination of the branches in the sequence. This is due to similar triangles. Suppose T R 3 L reaches greater height than T R 4 [See Figure 5], then the next comparison that would be made to determine tree height is height(t R 3 L 2 ) > height(t R 3 LR). The angles in the triangles used to find the last terms in the equations for height(t R 2 L) and height(t R 3 ) are congruent to the angle in the triangles we would use for height(t R 3 L 2 ) and height(t R 3 LR). Thus, if a right branch was chosen the first time these triangles were used in a comparison, then a right branch would be chosen again. This also means, any time a left branch is chosen, the path for greatest height in the tree is alternating from that point on (e.g. T R n (LR) ). Up to this point we have made the assumption that there exists a positive integer n for which r 2 cos((n 2) ) > cos(n ) is true but it is not necessarily clear that this is always the case. The following theorem will clarify this and provide an expression for n as a function of, r 2, and.

9 GEOMETRY OF ASYMMETRIC BINARY BRANCHING TREES 9 Figure 5 Theorem 3.. Suppose that 0 <, r 2 <, > r 2, and 0 < apple. Then, there 2 exists n 2 such that r 2 cos((n 2) ) cos(n ). Proof. Using trigonometric identities, the inequality r 2 cos((n 2) ) cos(n ) can be rewritten in the form: r 2 cos((n 2) ) (cos 2 cos((n 2) ) sin 2 sin((n 2) )) () r 2 cos((n 2) ) cos 2 cos((n 2) ) sin 2 sin((n 2) ) () cos((n 2) )(r 2 cos 2 ) sin 2 sin((n 2) ) () sin 2 sin((n 2) ) cos((n 2) )(r 2 cos 2 ) sin((n 2) ) cos 2 r 2 () cos((n 2) ) sin 2 r 2 () tan((n 2) ) cot 2 csc 2 () (n 2) arctan(cot 2 () n arctan(cot 2 r 2 csc 2 ) r 2 csc 2 ) + 2 Since arctan(cot 2 r 2 csc 2 ) + 2 is a constant, there is a smallest n such that r arctan(cot 2 2 r n csc 2 ) + 2. Now, we can write n as a function of, r 2, and :

10 0 A. O HANLON, P. HOWARD, AND D. BROWN n(, r 2, ) = p arctan(cot 2 r 2 csc 2 ) + 2q. By geometric series and summing all the terms for the heights of each right branch up to (and not including) n we get the following formula for tree height: (3.5) Xn (r j cos(j )) + n cos((n 2) ) + r n cos((n ) ). r j=0 If at the (n ) th branch the left and right branches are equal in height, there is more than one path determining tree height. From the first comparison in which the left and right branches were equal in height we know there must be at least two paths for tree height. Again, due to similar triangles, the height of the right and left branches will be equal at alternate branchings. Thus, if there is more than one path determining tree height, there are an infinitely many paths determining tree height. 4. Computer Implementation To generate pictures(footnote think of them on xy-coordinate plane) of binary branching fractal trees, a ne transformations are necessary. The two functions for symmetric trees are as follows: S R (x, y) = (rx cos( ) ry sin( ), rx sin( ) + ry cos( ) + ) S L (x, y) = (rx cos( ) ry sin( ), rx sin( ) + ry cos( ) + ) Since asymmetric binary branching trees have three variables involved in their construction it is quite di cult to visualize how minor changes in these variables may e ect the characteristics in the tree. For this reason, it is important to have a means of producing accurate pictures of these trees. As we discussed before, to generate these pictures a ne transformations are necessary. These transformations are: A L (x, y) = (r 2 x cos( ) r 2 y sin( ), r 2 x sin( ) + r 2 y cos( ) + ) A R (x, y) = ( x cos( ) y sin( ), x sin( ) + y cos( ) + ). Figure 6

11 GEOMETRY OF ASYMMETRIC BINARY BRANCHING TREES In Mandelbrot and Frame s article one of the main topics was self-contact in symmetric binary branching trees. In particular they discussed trees with = 35 and = 90. At = 35 the tree fills a right isosceles triangle [see Figure 6] and at = 90 the tree fills a rectangle [see Figure 7] with certain values of r. One of the characteristics of asymmetric trees that computer generated pictures allowed us to see was that these trees no longer completely fill their respective shapes. Furthermore, there will always be a gap in the lower left corner of the rectangle and on the left side of the trunk in the triangle, since > r 2. Figure 7 5. Conjectures There are many other topics concerning asymmetric branching trees that we have yet to study. For example in the previous section we discussed the fact that at = 90 and = 35 a symmetric branching tree can fill a rectangle or triangle, respectively, but with these same values of a gap is created in an asymmetric branching tree. One of the topics that we will be studying in the future is whether it is possible to measure this gap. Another topic that we will be considering is how to calculate tree height when 90 < < 80. In addition to these topics there are others that we have no immediate intention to study. One of these topics is a continuation of the topic of path length. For example, is it possible to find a pair of, and thereby infinitely many, equal paths within any asymmetric branching tree. Some of the characteristics of symmetric branching trees discussed by Mandelbrot and Frame, such as self contact and the shortest path in the canopy, can also be studied in asymmetric branching trees. [MF] References B. Mandelbrot and M. Frame, The canopy and shortest path in a self-contacting fractal tree, Math Intelligencer 2 (999), 2. A. O Hanlon, Department of Mathematics and Computer Science, Ithaca College, Ithaca, NY 4850, USA address: aohanlon@ic3.ithaca.edu

12 2 A. O HANLON, P. HOWARD, AND D. BROWN P. Howard, Department of Mathematics and Computer Science, Ithaca College, Ithaca, NY 4850, USA address: phoward@ic3.ithaca.edu D. Brown, Department of Mathematics, Ithaca College, Ithaca, NY , USA address: dabrown@ithaca.edu