Protected points in ordered trees

Transcription

1 Applied Mathematics Letters Protected poits i ordered trees Gi-Sag Cheo a, Louis W. Shapiro b, a Departmet of Mathematics, Sugkyukwa Uiversity, Suwo , Republic of Korea b Departmet of Mathematics, Howard Uiversity, Washigto, DC 0059, USA Received 9 Jue 007; accepted 0 July 007 Abstract I this ote we start by computig the average umber of protected poits i all ordered trees with edges. This ca serve as a guide i various orgaiatioal schemes where it may be desirable to have a large or small umber of protected poits. We will also look a few subclasses with a view to icreasig or decreasig the proportio of protected poits. c 007 Elsevier Ltd. All rights reserved. Keywords: Ordered tree; Protected poit; Catala umber; Fie umber; Cetral biomial coefficiet; Motki umbers; {0,, }-trees. Notatio ad overview A ordered tree [] is defied recursively. It is a tree with a root ad a ordered list of subtrees at the root. For istace the subtrees could be ordered by the time of creatio. The five ordered trees with three edges are show i Fig.. Fig.. Ordered trees with three edges. A protected poit is a vertex which is ot a leaf ad which is ot distace from a leaf. The root is ot cosidered to be a leaf except for the tree cosistig of oly the root. For istace, if leaves represet customers it may be worthwhile for may of the poits i the tree to be uprotected. However if the leaves represet lobbyists or computer hackers it may be a very good thig to have may poits protected. We will show that as the umber of edges gets large the average proportio of protected poits i all ordered trees approaches /6. The tool we will use is geeratig fuctios. A reasoable variatio occurs if we have a orgaiatioal tree such that the maximum umber of employees directly uder ay oe maager is at most two. Correspodig author. address: lou.shapiro@gmail.com L.W. Shapiro /$ - see frot matter c 007 Elsevier Ltd. All rights reserved. doi:0.06/j.aml

2 G.-S. Cheo, L.W. Shapiro / Applied Mathematics Letters If the out degree of ay poit is at most two the we are lookig at Motki trees. The same tools ca be used to show that the proportio of protected poits i Motki trees approaches 0/7. We look at these two cases i some detail ad the metio three more cases. Two geeratig fuctios which will use are those for the Catala umbers ad for the cetral biomial coefficiets. They are C = C = 4 = + C = C = 0 +, ad B = B = 4 = + C B = C = 0. A excellet source for backgroud iformatio o geeratig fuctios, Catala umbers ad Motki umbers is Staley s book [6]. For iformatio o Fie umbers see [,3] while asymptotics are discussed i [] ad somewhat similar applicatios of tree structure are discussed i [4,5]. It is well kow that the umber of ordered trees o edges is the th Catala umber, C = +. We use the terms poit ad vertex iterchageably. The umber of vertices i ay tree with edges is + so the geeratig fuctio for the umber of ordered trees with a distiguished poit is B = Alteratively this couts all vertices i all ordered trees. If we cout leaves which are vertices of up degree 0 the we fid the umbers,, 3, 0, 35, 6,... which suggests the geeratig fuctio B + /. The other geeratig fuctio we will eed is that for the umber of trees where the root is a protected poit or is the empty tree. Tryig small cases gives the umbers, 0,,, 6, 8, 57,.... I Fig. where = 3 we see that the two trees o the left have protected roots. Sice each subtree out of the root must have oe edge coectig to the root the geeratig fuctio for a sigle edge is ad a otrivial tree attached to this edge with geeratig fuctio C, each subtree at the root cotributes C = C ad the total geeratig fuctio is + C + C + C 3 + = C. This sequece of umbers is called the Fie umber sequece [3] ad the geeratig fuctio for the sequece is deoted as F = F = C = C + C = C + C. For a good referece for this material see [6], ad for asymptotic estimates the followig lemma of Beder is easy to apply ad very useful. Theorem. Beder s Lemma []. Suppose that A = 0 a ad B = 0 b are two geeratig fuctios, ad the radius of covergece of A is larger tha that of B. Let C = 0 c be the product A B. Suppose further that b /b approaches a limit b as. If A b 0, the c A b b.. The mai result I ordered trees ad i similar classes of trees the followig observatio holds: V = LT where V is the geeratig fuctio for trees with a distiguished vertex, L is the geeratig fuctio for trees with a distiguished leaf ad T is the geeratig fuctio for the umber of trees i the class. The way to see that this holds

3 58 G.-S. Cheo, L.W. Shapiro / Applied Mathematics Letters is to sip the distiguished vertex i half. This produces two trees. The first is a tree ow with a distiguished leaf ad a secod which was the subtree growig up from the distiguished vertex. Theorem.. The average portio of protected poits i all ordered trees with edges approaches /6 as. Proof. For ordered trees we have T = C, V = B where C ad B are geeratig fuctios for the Catala umbers ad the cetral biomial coefficiets, respectively, ad what looks like L = B + /. Assumig this from we would have B = B + C or equivaletly B = CB +. To prove this we write the right had side as B + C = ad simplify. This does provide a proof that L = B + /. If we wat the distiguished poit to be protected we wat the tree o top to be otrivial ad to have its root protected. Thus the appropriate geeratig fuctio is F where F is the geeratig fuctio for the Fie umbers give by. Hece we have LF = B C C + C = B + C B C. After expressig C ad B i terms of 4 i this equatio we obtai LF = 4 + The first few terms of LF are O Now, we are ready to fid asymptotic values. Asymptotically the first two terms of the right had side i 3 are irrelevat ad usig Beder s lemma with A = +4, B = = 4 0 ad b = 4, we see that [ ] /4 /4 + 4 = 6 where [ ] is the coefficiet operator. Sice the total umber of poits is protected poits approaches /6. For umerical reassurace we ote that [ 50 ]LF [ 50 ]B, = = we have that the average umber of Does a system where each staff member ca hire at most two uderligs afford a higher percetage of protected poits? To determie this we look at {0,, }-trees where the out degree of every vertex is 0,, or. The umbers of these trees are couted by the Motki umbers M with the geeratig fuctio M = M = 3. It is also kow that the umber of all vertices i {0,, }-trees, i.e. the umber of {0,, }-trees with a distiguished vertex, has the geeratig fuctio V give by V = 0 + M = d d M 4

4 G.-S. Cheo, L.W. Shapiro / Applied Mathematics Letters sice ay tree with edges has + vertices. But the, after some maipulatio, V = d d M = 3. 3 Sice V = LT we see that the umber of {0,, }-trees with a distiguished leaf has the geeratig fuctio L = V T = 3. 5 We ote that L has a sigularity at = /3 so the radius of covergece about = 0 is also /3 ad the ratio test the tells us that l + lim = 3, where l = [ ]L. l To get our asymptotic result we ow express everythig i terms of l. Sice V may be rewritte as V = 3, asymptotically we have [ ] V = [ ] 3 = l + l + 6 9l 3l = 3l as. 7 This is a result of some idepedet iterest sice it tells us that for {0,, }-trees as gets large about /3 of the vertices are leaves. Theorem.. The average portio of protected poits i {0,, }-trees with edges approaches 0/7 as. Proof. By a similar argumet, the umber of {0,, }-trees where the root is a protected poit or the empty tree has the geeratig fuctio K := + M + M, where M is the geeratig fuctio for the Motki umbers give by 4. Thus, the geeratig fuctio for the umber of protected poits o {0,, }-trees is give by LK = LM + M = M = The first few terms of LK are O 0. Asymptotically the last two terms of the right had side i 8 are irrelevat ad [ ]LK = [ ] = l 4l l + + l + 9 l 4l 3l + 9l = 0 9 l as. 9

5 50 G.-S. Cheo, L.W. Shapiro / Applied Mathematics Letters From 7 ad 9, we ca ow estimate the average umber of protected poits i {0,, }-trees as : [ ]LK [ ] V We ote that [ 00 ]LK [ 00 ]V 0 9 l = 0 = l 7 = = Here are three more structures but we ow omit details sice the method is the same. Aother protocol might be that everyoe has two employees if they have ay at all. This situatio is modeled by. The ratio of protected complete biary trees ad we have i that case that T = C, L = B ad V = 0 poits to all poits is as. + Yet aother protocol is that everyoe ca hire a juior or a seior employee or both but the two positios are differet. The model is icomplete biary trees ad we have T = C, L = B ad V = + 0. This time the ratio of protected poits to all poits is = 9 as. 56 For our last example we cosider complete terary trees where the out degree of every vertex is 0 or 3. Usig similar methods we fid that the ratio of protected poits to all poits is as. This is a very small umber of protected poits. May variatios are possible usig the same tools ad the method is more importat tha ay particular case. The method may be thought of as takig four steps:. Fid the umber of protected poits at the root for the class of ordered trees beig cosidered.. Use the V = L T equatio to fid the geeratig fuctio L. 3. Use L to trasport the geeratig fuctio at the root to a arbitrary vertex. 4. Fid the asymptotic value, ofte by usig Beder s lemma or at least the radius of covergece of the relevat geeratig fuctios. Ackowledgmets We would like to thak the referee for thoughtful ad helpful suggestios. The secod author was supported i part by NSF Grat HRD Refereces [] E. Beder, Asymptotic methods i eumeratio, SIAM Rev [] E. Deutsch, L.W. Shapiro, A bijectio betwee ordered trees ad -Motki paths ad its may cosequeces, Discrete Math [3] E. Deutsch, L.W. Shapiro, A survey of the Fie umbers, Discrete Math [4] J.D. Farley, Breakig Al-Qaeda cells: A mathematical aalysis of couterterrorism operatios A guide for risk assessmet ad decisio makig, Stud. Coflict & Terrorism [5] R.J. Wilso, J.J. Watkis, Graphs, a Itroductory Approach, Wiley, 990, pp [6] R. Staley, Eumerative Combiatorics, vol., Cambridge Uiversity Press, 999 Chapter 6, Exercises 9 ad 38.