aa r X i v : . [ c s . F L ] A p r About Fibonacci trees. − I − Maurice
Margenstern
April 30, 2019
Abstract
In this first paper, we look at the following question: are the propertiesof the Fibonacci tree still true if we consider a finitely generated tree bythe same rules but rooted at a black node? The direct answer is no, butnew properties arise, a bit more complex than in the case of a tree rootedat a white node, but still of interest.
This paper investigates a question the author raised to himself a long time agobut he had no time to look at it. When he established the properties of thefinitely generated tree he called the Fibonacci tree, he wandered whether theproperties still hold if, keeping the generating rules, we apply them to a treerooted at a black node.In section 2, we remind the reader the definitions about the Fibonacci treeand its connection with two tilings of the hyperbolic plane we call pentagrid and heptagrid . In section 3, we consider the properties which rely on the rep-resentations of the positive integers as sums of distinct Fibonacci numbers andwe look at the question raised in the abstract. We show that the propertiesattached to the Fibonacci tree rooted at a white node are no more true for thesimilar tree rooted at a black node. However, other properties can be estab-lished, more complicate in that setting. We also show the connection of theFibonacci tree rooted at a black node with the pentagrid and the heptagrid. Insection 4 we consider another increasing sequence of positive numbers which wecall the golden sequence which can also be attached to the Fibonacci trees, boththe tree rooted at a white node and the other rooted at a black one. Again,the nice properties which connect the golden sequence with the Fibonacci treerooted at a white node are no more true when it is rooted at a black node.However, other properties occur, more complicate than the previous ones, butstill worth of interest.Section 5 concludes the paper with open questions regarding the generaliza-tion of these results. It might be the goal of other papers.1
The Fibonacci trees
In sub section 2.1, we remind the reader the definition of the Fibonacci treeas well as different variations about it which were investigated by the authorin [1]. In sub section 2.2, we remind him/her the numbering of the nodes andthe important properties of their Fibonacci representations, in particular theconnections of the representation of the sons of a node with the representationof that node. In sub section 2.3, we remind the connection of the Fibonacci treewith the pentagrid and with the heptagrid.
We call
Fibonacci tree the finitely generated tree with two kinds of nodes, black nodes and white ones whose generating rules are: B → BW and W → BW W . (1)When the root of the tree is a white, black node, we call such a Fibonaccitree a white , black respectively, Fibonacci tree . In this whole section, weconsider white Fibonacci trees only. Black Fibonacci trees will be studied inSections 3 and 4. The status of a node says whether it is black or white.The connection with Fibonacci numbers first appear when we count thenumber of nodes which lay at the same level of the tree. For a white Fibonaccitree, we have the following property:
Theorem [1] In a white Fibonacci tree, the level of the root being , thenumber of nodes on the level k of the tree is f k +1 where { f n } n ∈ N is the Fibonaccisequence with initial conditions f = f = 1 . As in [1], let us number the nodes starting from 1 given to the root andthen, from level to level and, on each level, from left to right. From now on,we identify a node with the number it receives in the just described way. It isknown that any positive integer n can be written as a sum of distinct Fibonaccinumbers, the terms of the Fibonacci sequence considered in Theorem 1: n = k X i =1 a i f i with a i ∈ { , } . (2) The a i digits which occur in (1) are not necessarily unique for a given n . Theycan be made unique by adding the following condition : for 0 < i < k , if a i = 1,then a i − = 0. If we write the a i ’s of (1) as a word a ..a k , the condition foruniqueness of the representation says that in that word, the pattern neveroccurs. We call code of n the unique word attached to n by (1) when the pattern is ruled out. We also say code of ν for the node of the white Fibonacci treewhose number is ν , and we write [ ν ] for the code of ν . Formula (2) allows us torestore ν from its code [ ν ]. We shall write ν = ([ ν ]). From Theorem 1, we get:2 orollary [1] In a white Fibonacci tree, the leftmost node of the level k hasthe number f k and has the code k − for positive k . The rightmost node onthe same level has the number f k − and has the code ( ) k . We can state the following property:
Theorem [1] In a white Fibonacci tree, for any node ν we have that amongits sons a single one has [ ν ]00 as its code. That son is called the preferredson of ν . If the node is black, its preferred son is its black son, if the node iswhite, its preferred son is its white son in between its black son and the otherwhite one. Figure 1 illustrates the properties stated in the theorem. The reader mayeasily check them. Figure 1
The white Fibonacci tree. Representation of the first three levelsof the tree. In blue, the white nodes, in red, the black ones. Above a node,its number, under it, its code. Note that the preferred son property is trueas stated in Theorem . The green edges connect a node to its sons, whilethe orange ones connect a node n to the leftmost son of the node n +1 whenthat latter node is present on the considered level. As mentioned in the introduction, the white Fibonacci tree is connected withthe pentagrid and the heptagrid, two tilings of the hyperbolic plane. The pen-tagrid is the tessellation { , } , which means that the tiling is generated by thereflection of a basic polygon in its sides and the recursive reflections of the im-ages in their sides, where the basic polygon is the regular convex pentagon withright angles. That polygon lives in the hyperbolic plane, not in the Euclideanone. Similarly, the heptagrid is the tessellation { , } which is generated in a3imilar way where the basic polygon is the regular convex heptagon with 2 π Figure 2
The tilings generated by the white Fibonacci tree. To left, thepentagrid, to right, the heptagrid.
Figure 3 illustrates how the white Fibonacci tree generates the consideredtilings. In both tilings, the tiles of a sector , can be put in bijection with thenodes of the tree. In the case of the pentagrid, such a sector is a quarter ofthe plane: it is delimited by two perpendicular half-lines stemming from thesame vertex V of a tile τ and passing through the other ends of the edges of τ sharing V . Figure 3
How the white Fibonacci trees generate the pentagrid and theheptagrid: each isolated sectors in the above figures is spanned by the whiteFibonacci tree. The colours of the tile show the Fibonacci structure: bluetiles are the black nodes, green and yellow tiles are the white ones. Yellowtiles are the second white son of a white node. η and those half-lines pass through the mid-points of twoconsecutive sides of a tile sharing V as a vertex, V being also an end of η . Thereader is referred to [2] for proofs of the just mentioned properties. Accordingly,as shown on the figure, five sectors allow us to locate tiles in the pentagrid andseven sectors allows us to perform the same thing in the heptagrid. From nowon, we call tile ν the tile attached to the node ν of such a Fibonacci tree weassume to be fixed once and for all. We also say that [ ν ] is the code of the tile ν .The preferred son property allows us to compute in linear time with respectto the code of a node ν the codes of the nodes attached to the tiles which sharea side with the tile ν . Such tiles are called the neighbours of ν . Theorem 2also allows us to compute in linear time with respect to [ ν ] a shortest path inthe tiling, leading from the tile ν to tile 1. So far, we mentioned the properties of the white Fibonacci tree. Let us look atthe following problem with the black Fibonacci tree. In Sub section 3.1, we lookat the analog of Theorem 1 and its Corollary 1. In Sub section 3.2, we investigatethe analog of Theorem 2. In Sub section 3.3, we look at the connection of theblack Fibonacci tree with the pentagrid and with the heptagrid.
We can prove an analog of Theorem 1:
Theorem In a black Fibonacci tree, the level of the root being , the numberof nodes on the level k of the tree is f k where { f n } n ∈ N is the Fibonacci sequencewith initial conditions f = f = 1 . Corollary [1] In a black Fibonacci tree, the rightmost node of the level k has the number f k +1 and has the code k for non negative k . The property stated in Theorem 3 was noted in a paper by Kenichi Morita,but we shall see in Section 4 that both Theorems 1 and 3 simply come fromthe fact that the same generating rules are applied to the trees and that theinitial levels contain 1 and 3 nodes for the white tree while they contain 1 and2 nodes for the black one. The latter properties are in some sense symmetric:in the white Fibonacci tree the levels are Fibonacci numbers with odd index sothat the cumulative sum up to the current level plus one is a Fibonacci numberwith even index and in the black Fibonacci tree the situation is opposite: thelevels give rise to Fibonacci numbers with even index whose cumulative sum isprecisely a Fibonacci number with odd index. It is the reason why in the whitetree, the Fibonacci numbers with even index occur at the head of a level why inthe black one the Fibonacci numbers with odd index occur at the tail of a level.5 .2 The black Fibonacci tree and a successor property
The next question is: is Theorem 2 true for the black node if we number itsnodes in the same way as in the white tree and if we attach to the nodes thecodes of their numbers? The answer is no if we stick to the same statement andit is yes if we change the question by asking whether there is a rule to define theconnection of a node ν with the one whose code is [ ν ]00. Such a situation canbe suspected by comparing Theorem 1 with Theorem 3 as well as Corollary 1with Corollary 2. Figure 4
The black Fibonacci tree. The same convention about coloursof the nodes and of the edges between nodes as in Figure is used. We cansee that the preferred son property as stated in Theorem is not true in thepresent setting. Figure 4 shows us a first fact: the preferred son property is not observedin the black Fibonacci tree. We can see that for any black node ν , none of itssons has the code [ ν ]00 . More other, for the white nodes ν whose code can bewritten [ µ ]01 , the code [ µ ]0100 is that of the leftmost son of the node ν +1.We can now state the property which holds in the black Fibonacci tree. Asthe preferred son property is no more observed with the exception of a few whitenodes, we say that the successor of the node ν is the node whose code is [ ν ]00 .Theorem 2 says that in a white node, the successor of a node occurs amongits sons with a precise rule depending on the colour of the node. In order toformulate the property, we need to define the end of the code of a node. Thelast two digits of the code of a node is either , or . In that latter case,we know that in fact we can write the last three digits as as the pattern is ruled out. With two kinds of nodes, this would give us six classes of nodes.In fact we have five of them only, which can be written b00 , b01 , w00 , w01 and w10 , which we call the types of the nodes. The type of a node mixes itsstatus with the ending of its code. Accordingly, we can see that the type b01 defines an empty class, a property we shall check. For a node ν , succ ( ν ) is thenumber of its successor and s r ( ν ), s ℓ ( ν ) is the number of its rightmost, leftmost6on respectively. Note that we always have: s r ( ν )+1 = s ℓ ( ν +1). (3) Theorem In the black Fibonacci tree, we have succ ( ν ) = s r ( ν )+1 if ν is ablack node or a white node of type w01 . For the types of white node w00 and w10 , we have succ ( ν ) = s r ( ν ) . We have six rules giving the types of the sonsof a node according to its type: b00 → b01 - w10b01 → b01 - w10w00 → b00 - w01 - w00w00 ∗ → b01 - w10 - w00 ∗ w01 → b00 - w01 - w10w10 → b00 - w01 - w00 where the type w00 ∗ indicates a node of the form f k +1 . Proof. We prove the theorem by induction on all the properties listed in it. Wenote that the properties are trivially observed on Figure 4 for the root and itssons. Accordingly, we assume the property to be true for all nodes µ < ν andwe check that it is true for at least ν and ν +1. Let k +1 be the level of ν , with k ≥ µ be its father which is thus on the level k .From the statement of the theorem, namely by the rules induced on thetypes, we can deduce the following succession for two nodes κ and κ +1 lying onthe same level of the tree. b00 , w01 − b01 , w10 − w00 , b01 − w10 , b00 − w01 , w00 − w10 , w00 − w01 , w10 (4)Below, Tables 1 and 2 allow us to check the correctness of the statement ofTheorem 4. In both tables, the following notations are used for nodes connectedwith µ or ν . The node µ is µ −
1. The node ν is ν −
1. The node λ is s ℓ ( ν ) and λ is the node λ −
1. According to (3), λ = s r ( ν ). In both table, we indicatethe black nodes by writing their number in blue.In both tables, we apply the induction hypothesis for all nodes κ with κ < ν .In particular, that applies to ν too. Accordingly, λ , which is always a whitenode is known. If we know the code of ν , we know that of λ , of λ and thenthose of λ +1, λ +2 and λ +3.Indeed, if we know the code of κ we can easily compute the node of κ − κ +1. In [ κ ], say that the letter b is before the letter a if b is on the lefthand side of a . Say that a is safe if the letter before it is a too. Note thataccording to (2), the leftmost letter of a code is . We assume that there is asafe before the leading of the code. Lemma Let [ κ ] be the code of a node of a Fibonacci tree. Let [ κ ] = [ ξ ] ( ) n .Then, [ κ +1] = [ ξ ] ( ) n − . [ κ ] = [ ξ ] ( ) n . Then, [ κ +1] = [ ξ ] ( ) n .We can say that a pattern generates a carry which propagates until a safe which it replaces by a . emma Let [ κ ] be the code of a node of a Fibonacci tree. If [ κ ] ends with a , [ κ − can be written by replacing the rightmost by a . If [ κ ] = [ ξ ]( ) n .Then, [ κ −
1] = [ ξ ] ( ) n − . If [ κ ] = [ ξ ] ( ) n . Then, [ κ +1] = [ ξ ] ( ) n . Wecan say that a pattern generates or depending on the parity of therepetitions of the pattern before the rightmost . If that number is even itgenerates for each pattern ; otherwise, it generates the pattern . Table 1
Table of the computation of the codes of the following nodes when ν is a black node: µ is the father of ν , µ = µ − , ν = ν − , λ , the leftmostson of ν , λ = λ − , as well as the nodes λ +1 and λ +2 . We mark the blacknodes in blue, except when λ +2 is the successor of ν . µ µ ν ν λ λ λ +1 λ +2 µ is a black node, so that µ is white: α α β α β β β α β α β β β β β β µ is a black node, so that µ is white: β α β β β β β β β β β β β β β β µ and µ are both white nodes: α α α α α α α α α α α α α α α α Our proof consists in carefully looking at every possible case. We first assumethat ν is a black node. Its father µ may be white or black. Assume that µ − µ is white. Now, from the hypothesis, the possible types for µ and µ , taking into account that µ = µ +1, are : b00 , w01 or b01 , w10 . As µ < ν , the induction hypothesis applies to µ and to µ , From (4), we have that µ − w10 . Accordingly, its rightmost son ends in , so thatas mu is a black node, it is easy to check that ν ends in . If µ is black, sothat µ is white, the possible succession of the types are w00 , b01 and w10 , b00 :from the assumption, the rightmost son of a node is either w10 or w00 . If µ and µ are both white, then the only possibilities for the succession of types are w01 , w00 and w01 , w10 . Note that when µ is black or has the type w01 ,its successor is ν , otherwise it is ν . Similarly, if ν , which is always white hasthe type w01 , then its successor is λ , otherwise it is λ . From these featuresand the help of Lemmas 1 and 2 as well as (4), we obtain the computations ofTable 1. Note that in the application of 2, we have to take into account on theassumption hypothesis and on the change from β
01 to α
00, for instance, in orderto compute the code of κ − κ ends with several contiguous .Let us apply the similar arguments for the case when ν is a white node. Thistime, the computation is also based on the position of ν among the other sons ofits father µ . Consider the case when µ is black. Necessarily, µ is white and ν isblack. The possible types for µ , µ and ν are w10 , b00 , b01 and w00 , b01 , b01 .In those cases, the successor of µ is ν = ν −
1. The computation from ν to ν
8s straightforward from Lemma 1, so that we do not mention the code of ν inTable 2. Also, as ν has the type b01 , its successor is λ . Table 2
Table of the computation of the codes of the son of ν when it is awhite node whose father is µ . The same conventions as in Table are usedhere too. In the upper part of the table, µ is a black node, so that µ is white.In the lower part of the table, µ is a white node and we assume that ν toois a white node. In the upper part of the table, ν is black. That node maybe also white in the lower part of the table. ν is black, µ is white and is black: µ µ ν ν λ λ +1 λ +2 λ +3 β α β β β β β β α α α α β β β β ν and µ are black, µ is white: µ µ ν ν λ λ +1 λ +2 λ +3 α α α α α α α α α α α α α α α α ν is black, µ and µ are both white: µ µ ν ν λ λ +1 λ +2 λ +3 α α α α α α α α β α β β β β β β β α β β β β β β ν is a white node, so that µ is white too: µ µ ν ν λ λ +1 λ +2 λ +3 α α α α α α α α α α α α β β α α β α β α β β α α α α α α α α α α β α β α β β α α Consider still the case when ν is black. We have the case when µ is whiteand both µ is black. The possible types are b01 , w10 , b00 and b00 , w01 , b00 for the nodes µ , µ and ν in that order. Next, we have the case when both µ and µ are white nodes. For µ and µ the succession of types is w01 , w10 or w01 , w00 or also w10 , w00 . In the first two cases, the type of ν is b00 , whilein the exceptional case w10 , w00 , it is b01 .Now, consider the case when ν is white. Necessarily, µ is also a white node.When µ is black, we have the possibilities b01 , w10 and b00 , w01 . When µ is white, we have three possibilities: w01 , w00 or w01 , w10 or also w01 , w00 .This latter case happens when µ and ν are the last nodes of two consecutivelevels. We have already met this case when µ is the last node of a level and ν is the penultimate node on the next level. In that case ν is black: see Table 2.The last column of the tables shows the types of the sons of ν with the type9ritten in blue for the black nodes, which allows us to drop the letter of thestatus. We can see that the hypothesis is checked for the sons of the node ν andtheir types. This completes the proof of the theorem. (cid:3) It is time to indicate which place a black Fibonacci tree takes in the pentagridand in the heptagrid.As illustrated by Figure 5, the sectors defined by Figure 3 in Sub section 2.3can be split with the help of regions of the tiling generated by the white Fi-bonacci tree and by the black one.
S T
Figure 5
The decomposition of a sector spanned by the white Fibonaccitree into a tile, then two copies of the same sector and a strip spanned bythe black Fibonacci tree. To left: the decomposition in the pentagrid; toright, the decomposition in the heptagrid. In both cases, the lines whichdelimit a sector spanned by each kind of tree.
In the figure, the sector is split into a tile, we call it the leading tile , anda complement which can be split into two copies of the sector and a regionspanned by the black Fibonacci tree which we call a strip .In both tilings, the strip appears as a region delimited by two lines ℓ and ℓ which are non-secant. It means that they never meet and that they also are notparallel, a property which is specific of the hyperbolic plane. There is a thirdline which supports the side of the tile τ which is associated with the root of theblack Fibonacci tree. That line is the common perpendicular to ℓ and ℓ . Thetile τ is called the leading tile of the strip. It is worth noticing that the waywe used to split the sector can be recursively repeated in each sector generatedby the process of splitting. We can note that the strip itself can be exactlysplit into a tile, a sector and a strip. This process is closely related with thegenerating rules of the Fibonacci trees. At this point, it can be noticed thatthere are several ways to split a sector and a strip again into strips ans sectors.This can be associated with other rules for generating a tree which we againcall a Fibonacci tree. There are still two kinds of nodes, white and black ones.10ut the rules are different by the order in which the black son occurs amongthe sons of a node. There are two choices for black nodes and three choices forwhite ones. Accordingly, there are six possible definitions of Fibonacci tree. Wecan also decide to choose which rule is applied each time a node is met. In [1]those possibilities are investigated. We refer the interested reader to that paper. Figure 6
The decomposition of a sector spanned by the white Fibonaccitree into a sequence of pairwise adjacent strips spanned by the black Fi-bonacci tree. To left: the decomposition in the pentagrid; to right, thedecomposition in the heptagrid. In both cases, the lines which delimit thestrips spanned by the tree.
Figure 7
Another look on the decomposition of the sector given by Fig-ure . The structure of the Fibonacci is erased in order to highlight thedecomposition into pairwise adjacent strips. But a sector can be split in another way which is illustrated by figure 6.Consider a sector S . Consider its leading tile T . That tile is associated withthe root of the white Fibonacci tree. assume that we associate it with the blackFibonacci tree in such a way that in the association the leftmost son of T isagain the black son of the root in both trees. What remains in the sector? Itremains a node which we can associate with the root of the white Fibonacci tree.A simple counting argument, taking into account that the levels are different11y one step from the white tree to the black one in that construction, showsus that in this way we define an exact splitting of the sector. And so, there isanother way to split the sector: into a strip B and a sector again, S . Now,what was performed for S can be repeated for S which generates a strip B and a new sector S . Accordingly,we proved: Theorem The sector associated to the white Fibonacci tree can be split into asequence of pairwise adjacent strips B n , n ∈ N , associated to the black Fibonaccitree. Equivalently, the white Fibonacci tree can be split into the union of asequence of copies of the black Fibonacci tree. The leading tiles of the B n ’s areassociated with the nodes f n +1 − of the white Fibonacci tree, i.e. the nodeswhich are on the rightmost branch of the white Fibonacci tree. Let us go back to the generating rules of the white Fibonacci tree. From therules defined by (1) in Sub section 2.1, we can easily count the number of nodeswhich lie at the same level of the tree. Denote by u n , v n the number of white,black nodes respectively lying on the level n . Denote the total number of nodeson that level by w n . Clearly: u n +1 = 2 u n + v n v n +1 = u n from which we easily get: w n +2 = 3 w n +1 − w n . (5)Note that (5) can be found directly: white nodes generate three nodes andblack ones two nodes only, but the number of black nodes of the considered levelis the number of nodes of the previous level as for each node there is a singleblack son. Now, equation (5) defines a polynomial P ( X ) = X − X + 1 whoseroots are the numbers 3 + √
52 and 3 − √
52 . Now, 3 + √
52 = ( 1 + √
52 ) whichexplains the link with the Fibonacci sequence and why the number of nodesin the white Fibonacci tree are connected with the Fibonacci numbers whoseindex is odd.We can define codes with the sequence w n . Indeed, it is not difficult to provethat any positive number n is a sum of distinct terms of the sequence { w n } n ∈ N + ,where N + is the set of positive integers. We have: n = k X j =1 a j w j with a j ∈ { , , } , (6)where the sequence { w n } n ∈ N + is defined by (5) and the initial conditions w = 1with w = 3 for the white Fibonacci tree and w = 1 with w = 2 for the blackFibonacci tree. As in the case of the representation of positive numbers Fi-bonacci, this representation is not unique. Also, we can associate to the a j ’s of12ormula (6) a word in the alphabet { , , } which we write a k ...a and whichwe call the golden code of n and we write again [ n ] when there is no ambiguitywith the code defined with the Fibonacci numbers, otherwise we write [ n ] g forthe golden code. We can made the golden code of n unique by requiring thatthe pattern ∗ is ruled out, where ∗ is either the empty word or a wordconsisting of ’s only. The code associated with the Fibonacci numbers willhere be called Fibonacci code .We can also associate the golden code to a tile in a fixed sector of thepentagrid or of the heptagrid by giving to the tile the golden code of the numberassociated to the tile.In Sub section 4.1, we look at the properties of the golden code in the whiteFibonacci tree. In Sub section 4.2, we look at the same issue in the blackFibonacci tree. We shall see an analogous phenomenon with what we observedin the previous sections.
The golden codes in the white Fibonacci tree have properties which are similarto those of the Fibonacci codes we have depicted in Sub section 2.2 in the sametree.The author and its co-author, Gencho Skordev, proved that the preferred sonproperty is true with the golden codes. However, due to the different alphabetused for writing the codes, the definition of the preferred son in this context isa bit different. The definition comes with the following result:
Theorem (see [3, 2]) In the white Fibonacci tree fitted with the golden codes,each node has exactly one node among its sons whose code ends with . Thatson is called the preferred son of the node. Moreover, if [ ν ] is the golden codeof ν , the golden code of its preferred son is [ nu ] . In all nodes, the preferredson is the leftmost white son. Figure 8
The golden codes in the white Fibonacci tree: we can check theproperties stated by Theorem .
13e refer the reader to [3, 2] for the proof of the result which is stated forthe general case in the quoted references.
In this context too, when we look at the golden codes in the black Fibonaccitree, we can see that the preferred son property is no more true, as it can easilybe checked on Figure 9.However, there is a kind of regularity, more regular than in the case of theFibonacci codes, which are indicated in Theorem 7. As in Sub section 3.2, wecall successor of the node ν , the node whose golden code is [ ν ] g , the nodebeing again denoted by succ ( ν ). Theorem In the black Fibonacci tree, for all nodes ν , we have that, exceptfor the white nodes whose golden code ends in , succ ( ν ) = s r ( ν )+1 . For thewhite nodes whose golden code ends in , we have succ ( ν ) = s r ( ν ) . Moreover,the last two letters of a golden codes combined with the statuses of the nodesgive rise to five combinations: b0 , b1 , w0 , w1 and w2 . Each type give risesto a specific rule determining the types of the sons: b0 → b0 , w1b1 → b1 , w2w0 → b0 , w1 , w0w1 → b0 , w1 , w2w2 → b0 , w1 , w2 The proof is very similar in its principle to the proof of Theorem 4. This iswhy, here, it will boil down to Tables 4 and 3. We just append the followingremark: the successive types of two consecutive nodes are the following ones,which is a consequence of the rules and which are used also among the inductionhypothesis: b0 , w1 − b1 , w2 − w1 , b0 − w2 , b0 − w1 , w0 − w1 , w2 (7)Indeed, we shall check by induction that the case w0 , b1 never occurs fornodes on the same level as a second white son in occurs only for the last nodeof a level.Before turning to the tables, we can check the properties stated in Theorem 7in Figure 9. In the tables, the notations for the numbers of the nodes are thesame as in Tables 1 and 2. We have again, that λ = s ℓ ( ν ) = s r ( ν ), and that ν = s r ( µ ).Although the general splitting of the proof is the same as in the case of theFibonacci codes, there are in this case specificities connected with the specialforbidden pattern ∗ and with the fact that a node of type w0 has its rightmostson as its successor. We also have to check that such a node occurs as the lastnode of the level only. Also, the succession of types for two nodes is ruledby (7), which is different from (4) in the codes based on Fibonacci numbers.14he situation of the white node ending with appears in Table 4 as the thirdline of the case when ν is also white. We can see that a code β is followedby the code α , which means that β ends with a followed by ’s only andso that α is the successor of β . This explains why the code of λ +1, β isfollowed by the code α which is that of λ +2 as expected from the statementof Theorem 7 as in this case, the type of ν is w0 . From this and from the table,we can see that a rightmost son of type w0 occurs at this line only, so that theassumption that the last node on a level is of that type is checked. Table 3
Table of the computation of the golden codes of the following nodes when ν is a black node. We mark the black nodes in blue, except when λ +2 is the successorof ν . µ µ ν ν λ λ λ +1 λ +2 µ is a black node, so that µ is white: α α β α β β β α α α β α β β β α α α α α α α α α µ is a white node and µ is black: β α β β β β β β β α β β β β β β µ and µ are white: β α β β β β β β α α α α α α α α Figure 9
The golden codes in the black Fibonacci tree: we can check theproperties stated by Theorem . Also note that in Table 3, the first two lines of the table indicates twodifferent situations with the predecessor of a code α . In the first line, we havethe standard situation where the predecessor is β . In the second line, thepredecessor is β . This means that in that case, β ends in a pattern ∗ , so15hat the next node on the level has the code α ∗ . Also note that when the codeof the node κ is β and that of κ +1 is α , it means that β ends with the pattern ∗ , so that the code β k is followed by α ∗ . In the same line, if the code of κ is α and that of κ +1 is α , it means that α does not end with ∗ , so thatthe code α k +1 is followed by α k . Table 4
Table of the computation of the golden codes of the following nodes when ν is a white node. We mark the black nodes in blue, except when λ +2 is the successorof ν . ν and µ are black nodes, so that µ is white: µ µ ν ν λ λ +1 λ +2 λ +3 β α β β β β β β β α β β β β β β ν is a black node, and µ is white: µ µ ν ν λ λ +1 λ +2 λ +3 α α α α α α α α α α α α α α α α β α β β β β β β α α α α α α α α ν is a white node, and so µ is white too: µ µ ν ν λ λ +1 λ +2 λ +3 α α α α α α α α α α α α α α α α β α β α β β α α α α α α α α α α With those last remarks and Table 4, we completed the proof of Theorem 7. (cid:3)
We can conclude the paper with several remarks.The first one is the interest of the golden representation, which is not usedas intensively as it should be by the author itself, although he found out theproperty stated by Theorem 6 a long time ago. The property stated in Theo-rem 7 is rather unexpected. It is a new one and it explains the weak use of thatencoding.The second remark, connected with both the Fibonacci and the golden rep-resentations is that the white Fibonacci tree is the best tree for navigationpurpose in the pentagrid and in the heptagrid, those tessellation that live in the16yperbolic plane.A third remark is that it was proved in [3, 2], that the properties found out inthe pentagrid and in the heptagrid can be generalized to the tessellations { p, } and { p +2 , } , still in the hyperbolic plane, where p ≥
5. As in the case of thepentagrid and of the heptagrid which corresponds to the case p = 5, for each p ,a specific tree generates the tiling in both { p, } and { p +2 , } . Interestingly, therules associated with such a tree extend in some sens the rules used to constructthe Fibonacci trees. And so, in that generalized context, the white and theblack trees also exist. Now, as the preferred son property is also true in thewhite tree, this time for an extension of the golden encoding, we can wonderwhether an extension of Theorem 7 is true for the black tree. References [1] M. Margenstern, New Tools for Cellular Automata of the Hyperbolic Plane,
Journal of Universal Computer Science , (12), (2000), 1226–1252.[2] M. Margenstern, Cellular Automata in Hyperbolic Spaces, vol. , Theory ,Collection: Advances in Unconventional Computing and Cellular Automata ,Editor: Andrew Adamatzky, Old City Publishing, Philadelphia, (2007),424p.[3] M. Margenstern, G. Skordev, Fibonacci Type Coding for the Regular Rect-angular Tilings of the Hyperbolic Plane,
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