A Brief History of Strahler Numbers

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1 A Brief History of Strhler Numbers Jvier Esprz, Mihel Luttenberger, nd Mximilin Shlund Fkultät für Informtik, Tehnishe Universität Münhen, Germny Abstrt. The Strhler number or Horton-Strhler number of tree, originlly introdued in geophysis, hs surprisingly rih theory. We sketh some milestones in its history, nd its onnetion to rithmeti expressions, grph trversing, deision problems for ontext-free lnguges, Prikh s theorem, nd Newton s proedure for pproximting zeros of differentible funtions. The Strhler Number In 945, the geophysiist Robert Horton found it useful to ssoite strem order to system of rivers (geophysiists seem to prefer the term strem ) [0]. Unbrnhed fingertip tributries re lwys designted s of order, tributries or strems of the d order reeive brnhes or tributries of the st order, but these only; 3d order strem must reeive one or more tributries of the d order but my lso reeive st order tributries. A 4th order strem reeives brnhes of the 3d nd usully lso of lower orders, nd so on. Severl yers lter, Arthur N. Strhler repled this mbiguous definition by simpler one, very esy to ompute [6]: The smllest, or finger-tip, hnnels onstitute the first-order segments. [... ]. A seond-order segment is formed by the juntion of ny two first-order strems; third-order segment is formed by the joining of ny two seond order strems, et. Strems of lower order joining higher order strem do not hnge the order of the higher strem. Thus, if first-order strem joins seond-order strem, it remins seond-order strem. Figure shows the Strhler number for frgment of the ourse of the Elbe river with some of its tributries. The strem system is of order 4. From omputer siene point of view, strem systems re just trees. Definition. Let t be tree with root r. The Strhler number of t, denoted by S(t), is indutively defined s follows. If r hs no hildren (i.e., t hs only one node), then S(t) = 0.

2 4 3 3 Fig.. Strhler numbers for frgment of the Elbe river. If r hs hildren r,..., r n, then let t,..., t n be the subtrees of t rooted t r,..., r n, nd let k = mx{s(t ),..., S(t n )}: if extly one of t,..., t n hs Strhler number k, then S(t) = k; otherwise, S(t) = k +. Note tht in this forml definition the Strhler number of simple hin ( finger-tip ) is zero, nd not one. This llows nother hrteriztion of the Strhler number of tree t s the height of the lrgest minor of t tht is perfet binry tree (i.e., rooted tree where every inner node hs two hildren nd ll leves hve the sme distne to the root): Roughly speking, suh binry tree is obtined by, strting t the root, following pths long whih the Strhler number never dereses by more thn one unit t time, nd then ontrting ll nodes with only one hild. If t itself is binry tree, then this minor is unique. We leve the detils s smll exerise. Figure shows trees with Strhler number,, nd 3, respetively. Eh node is lbeled with the Strhler number of the subtree rooted t it. Together with other prmeters, like bifurtion rtio nd men strem length, Horton nd Strhler used strem orders to derive quntittive empiril lws for strem systems. Tody, geophysiists spek of the Strhler number (or Horton-Strhler number) of strem system. Aording to the exellent Wikipedi rtile on the Strhler number (minly due to Dvid Eppstein), the Amzon nd the Mississippi hve Strhler numbers of 0 nd, respetively.

3 Fig.. Trees of Strhler number,, nd 3. Strhler Numbers nd Tree Trversl The first pperne of the Strhler number in Computer Siene seems to be due to Ershov in 958 [8], who observed tht the number of registers needed to evlute n rithmeti expression is given by the Strhler number of its syntx tree. For instne, the syntx tree of (x + y z) t, shown on the left of Figure 3, hs Strhler number, nd indeed n be omputed with just two registers R, R by mens of the ode shown on the right. + x y z w R y R z R R R R x R R + R R w R R R Fig. 3. An rithmeti expression of Strhler number The strtegy for evluting expression e = e op e is esy: strt with the subexpression whose tree hs lowest Strhler number, sy e ; store the result in register, sy R ; reuse ll other registers to evlute e ; store the result in R ; store the result of R opr in R. Ershov s observtion is relled by Fljolet, Roult nd Vuillemin in [6], where they dd nother observtion of their own: the Strhler number of binry tree is the miniml stk size required to trverse it. Let us tth to eh node of the tree the Strhler number of the subtree rooted t it. The trversing proedure follows gin the lowest-number-first poliy (notie tht rithmeti expressions yield binry trees). If node with number k hs two hildren, then

4 4 the trversing proedure moves to the hild with lowest number, nd pushes the (memory ddress of the) other hild onto the stk. If the node is lef, then the proedure pops the top node of the stk nd jumps to it. To prove tht the stk size never exeeds the Strhler number, we observe tht, if node of number k hs two hildren, then t lest one of its hildren hs number smller thn k. So the proedure only pushes node onto the stk when it moves to node of stritly smller number, nd we re done. Notie, however, tht the lowest-number-first poliy requires to know the Strhler number of the nodes. If these re unknown, ll we n sy is tht nondeterministi trversing proedure lwys needs stk of size t lest equl to the Strhler number, nd tht it my sueed in trversing the tree with stk of extly tht size.. Distribution of Strhler Numbers The gol of Fljolet, Roult nd Vuillemin s pper is to study the distribution of Strhler numbers in the binry trees with fixed number n of leves. Let S n be the rndom vrible orresponding to the Strhler number of binry tree (every node hs either two or 0 hildren) with n internl nodes hosen uniformly t rndom. Sine the Strhler number of t is the height of the lrgest perfet binry tree embeddble in t, we immeditely hve S n log (n+). The pper shows tht Exp[S n ] log 4 n nd Vr[S n ] O(). In other words, when n grows the Strhler number of most trees beomes inresingly loser to log 4 n. Independently of Fljolet, Roult nd Vuillemin, lso Kemp derives in [] the sme symptoti behviour of the expeted Strhler number of rndom binry tree. Lter, Fljolet nd Prodinger extend the nlysis to trees with both binry nd unry inner nodes [7]. Finlly, Devroye nd Kruszewski show in [5] tht the probbility tht the Strhler number of rndom binry tree with n nodes devites by t lest k from the expeted Strhler number of log 4 n is bounded from bove by, tht is, the Strhler number is 4 k highly onentrted round its expeted vlue.. Strhler Numbers in Lnguge Theory: Derivtion indies, Cterpillrs, nd Dimensions Derivtion indies nd terpillrs. The Strhler number hs been redisovered (multiple times!) by the forml lnguge ommunity. In [9], Ginsburg nd Spnier introdue the index of derivtion S α α w of given grmmr s the mximl number of vribles ourring in ny of the sententil forms α i (see lso [7]). For instne, onsider the grmmr X XX b. The index of the derivtions X XX XXX bxx bbx bbb X XX bxx bxx bbx bbb

5 5 is 3 (beuse of XXX) nd, respetively. For ontext-free grmmrs, where we hve the notion of derivtion tree of word, we define the index of derivtion tree s the miniml index of its derivtions. If the grmmr is in Chomsky norml form, then derivtion tree hs index k if nd only if its Strhler number is (k ). A first use of the Strhler number of derivtion trees n be found in [4], where Chytil nd Monien, pprently unwre of the Strhler number, introdue k-terpillrs s follows: A terpillr is n ordered tree in whih ll verties of outdegree greter thn one our on single pth from the root to lef. A -terpillr is simply terpillr nd for k > k-terpillr is tree obtined from terpillr by repling eh hir by tree whih is t most (k )- terpillr. Clerly, tree is k-terpillr if nd only if its Strhler number is equl to k. Let L k (G) be the subset of words of L(G) hving derivtion tree of Strhler number t most k (or, equivlently, being k-terpillr). Chytil nd Monien prove tht there exists nondeterministi Turing mhine with lnguge L(G) tht reognizes L k (G) in spe O(k log G ). Assume for simpliity tht G is in Chomsky norml form. In order to nondeterministilly reognize w =... n L k (G), we guess on-the-fly (i.e., while trversing it) derivtion tree of w with Strhler number t most k, using stk of height t most k. The trversing proedure follows the smller-number-first poliy. More preisely, the nodes of the tree re triples (X, i, j) with intended mening X genertes tree with yield i... j. We strt t node (S,, n). At generi node (X, i, j), we proeed s follows. If i = j, then we hek tht X i is prodution, pop new node, nd jump to it. If i < j, then we guess prodution, sy X Y Z, nd n index i l j, guess whih of (Y,, i) nd (Z, l, j) genertes the subtree of lowest number, sy (Y, i, l), nd jump to it, pushing (Z, l, j) onto the stk. The trversing proedure n lso be used to hek emptiness of L k (G) in nondeterministi logrithmi spe (remember: k is not prt of the input) [3]. In this se we do not even need to guess indies: if the urrent node is lbeled by X, then we proeed s follows. If X hs no produtions, then we stop. If G hs prodution X for some terminl, we pop the next node from the stk nd jump to it. If G hs produtions for X, but only of the form X Y Z, then we guess one of them nd proeed s bove. Notie tht heking emptiness of L(G) is P -omplete problem, nd so unlikely to be solvble in logrithmi spe. Tree dimension. The uthors of this pper re lso guilty of redisovering the Strhler number. In [9] we defined the dimension of tree, whih is... nothing but its Strhler number. Severl ppers [9,, 8, 3] hve used tree dimension The nme dimension ws hosen to reflet tht trees with Strhler number re hin (with hirs), trees of dimension re hins of hins (with hirs), tht n be

6 6 (tht is, they hve used the Strhler number) to show tht L n+ (G), where n is the number of vribles of grmmr G in Chomsky norml form, hs interesting properties : () Every w L(G) is sttered subword of some w L n+ (G) [3]. () For every w L(G) there exists w L n+ (G) suh tht w nd w hve the sme Prikh imge, where the Prikh imge of word w is the funtion Σ N tht ssigns to every terminl the number of times it ours in w. Equivlently, w nd w hve the sme Prikh imge if w n be obtined from w by reordering its letters [9]. The first property hs lredy found t lest one interesting pplition in the theory of forml verifition (see [3]). The seond property hs been used in [] to provide simple onstrutive proof of Prikh s theorem. Prikh s theorem sttes tht for every ontext-free lnguge L there is regulr lnguge L suh tht L nd L hve the sme Prikh imge (i.e., the set of Prikh imges of the words of L nd L oinide). For instne, if L = { n b n n 0}, then we n tke L = (b). The proof desribes proedure to onstrut this utomton. By property (), it suffies to onstrut n utomton A suh tht L(A) nd L k+ (G) hve the sme Prikh imge. We onstrut A so tht its runs simulte the derivtions of G of index t most k +. Consider for instne the ontext-free grmmr with vribles A, A (nd so k = ), terminls, b,, xiom A, nd produtions A A A A ba A A Figure 4 shows on the left derivtion of index 3, nd on the right the run of A simulting it. The sttes store the urrent number of ourrenes of A nd A, nd the trnsitions keep trk of the terminls generted t eh derivtion step. The run of A genertes b, whih hs the sme Prikh imge s b. The omplete utomton is shown in Figure 5. 3 Strhler Numbers nd Newton s method Finlly, we present surprising onnetion between the Strhler number nd Newton s method to numerilly pproximte zero of funtion. The onnetion works for multivrite funtions, but in this note we just onsider the univrite se. Consider n eqution of the form X = f(x), where f(x) is polynomil with nonnegtive rel oeffiients. Sine the right-hnd-side is monotoni funtion, by Knster-Trski s or Kleene s theorem the eqution hs extly one smllest niely drwn in the plne, trees of dimension 3 re hins of hins of hins (with hirs), with n be niely displyed in 3-dimensionl spe, et. For n rbitrry grmmr G, the sme properties hold for L nm+(g), where m is the mximl number of vribles on the right-hnd-side of prodution, minus. If G is in Chomsky norml form, then m.

7 7 A (0, ) A A ɛ (, ) A ba A A ba A ba A ba ba b b (, ) (, ) (, ) (0, ) (, 0) (0, 0) Fig. 4. A derivtion nd its simultion. 0, 0, 0, 0 3, 0 ε ε 0,,, b ε b 0,, b 0, 3 Fig. 5. The Prikh utomton of A A A, A ba A A with xiom A. solution (possibly equl to ). We denote this solution by µf. It is perhps less known tht µf n be given lnguge-theoreti interprettion. We explin this by mens of n exmple (see [4] for more detils). Consider the eqution X = 4 X + 4 X + () It is equivlent to (X )(X ) = 0, nd so its lest solution is X =. We introdue identifiers, b, for the oeffiients, yielding the forml eqution X = f(x) := X + bx +. () We rewrite this eqution s ontext-free grmmr in Greibh norml form in the wy one would expet: G : X XX bx, (3)

8 8 Consider now the derivtion trees of this grmmr. It is onvenient to rewrite the derivtion trees s shown in Figure 6: We write terminl not t lef, but t its prent node, nd so we now write the derivtion tree on the left of the figure in the wy shown on the right. Notie tht, sine eh prodution genertes different terminl, both representtions ontin extly the sme informtion. 3 X X X b b X X X b b X Fig. 6. New onvention for writing derivtion trees We ssign to eh derivtion tree t its vlue V (t), defined s the produt of the oeffiients lbeling the nodes. So, for instne, for the tree of Figure 6 we get the vlue b 3 = (/4) 4 (/) 3 = /8. Further, we define the vlue V (T ) of set T of trees s t T V (t) (whih n be shown to be well defined, even if T is infinite). If we denote by T G the set of ll derivtion trees of G, then µf = V (T G ). (4) The erliest referene for the this theorem in ll its generlity we re wre of is Bozplidis [] (Theorem 6) to whom lso [6] gives redit. A well-known tehnique to pproximte µf is Kleene itertion, whih onsists of omputing the sequene {κ i } i N of Kleene pproximnts given by κ 0 = 0 κ i+ = f(κ i ) for every i 0 It is esy to show tht this orresponds to evluting the derivtion trees (with our new onvention) by height. More preisely, if H i is the set of derivtion trees of T G of height less thn i, we get κ i = V (H i ) (5) In other words, the Kleene pproximnts orrespond to evluting the derivtion trees of G by inresing height. 3 This little hnge is neessry, beuse the tree of the derivtion X hs Strhler number if trees re drwn in the stndrd wy, nd 0 ording to our new onvention.

9 9 It is well known tht onvergene of Kleene itertion n be slow: in the worst se, the number of orret digits grows only logrithmilly in the number of itertions. Newton itertion hs muh fster onvergene (f. [5, 0, 5]). Rell tht Newton itertion pproximtes zero of differentible funtion g(x). For this, given n pproximtion ν i of the zero, one geometrilly omputes the next pproximtion s follows: ompute the tngent to g(x) t the point (ν i, g(ν i )); tke for ν i+ the X-omponents of the intersetion point of the tngent nd the x-xis. For funtions of the form g(x) = f(x) X, n elementry lultion yields the sequene {ν i } i N of Newton pproximnts ν 0 = 0 ν i+ = ν i f(ν i) ν i f (ν i ) We remrk tht in generl hoosing ν 0 = 0 s the initil pproximtion my not led to onvergene only in the speil ses of the nonnegtive rels or, more generlly, ω-ontinuous semirings, onvergene is gurnteed for ν 0 = 0. A result of [] (lso derived independently in [3]) shows tht, if S i is the set of derivtion trees of T G of Strhler number less thn i (where trees re drwn ording to our new onvention), then ν i = V (S i ) (6) In other words, the Newton pproximnts orrespond to evluting the derivtion trees of G by inresing Strhler number! The onnetion between Newton pproximnts nd Strhler numbers hs severl interesting onsequenes. In prtiulr, one n use results on the onvergene speed of Newton itertion [3] to derive informtion on the distribution of the Strhler number in rndomly generted trees. Consider for instne rndom trees generted ording to the following rule. A node hs three hildren with probbility 0., two hildren with probbility 0., one hild with probbility 0., nd zero hildren with probbility 0.6. Let G the ontext-free grmmr X XXX bxx X d with vlution V () = 0., V (b) = 0., V () = 0., V (d) = 0.6. It is esy to see tht the probbility of generting tree t is equl to its vlue V (t). For instne, the tree t of Figure 7 stisfies Pr[t] = V (t) = b d 5. Therefore, the Newton pproximnts of the eqution X = 0.X X + 0.X + 0.6

10 0 b d d d b d d Fig. 7. A tree with probbility b d 5 give the distribution of the rndom vrible S tht ssigns to eh tree its Strhler number. Sine f(x) = 0.X 3 +0.X +0.X +0.6 nd f (X) = 0.3X + 0.4X + 0., we get ν 0 = 0.6 ν i+ = ν i ν3 i + ν i 9ν i + 6 3ν i + 4ν i 9 nd so for the first pproximnts we esily obtin ν 0 = Pr[S < 0] = 0 ν = Pr[S < ] = ν = Pr[S < ] ν 3 = Pr[S < 3] ν 4 = Pr[S < 4] As we n see, the probbility onverges very rpidly towrds. This is not oinidene. The funtion f(x) stisfies µf <, nd theorem of [3] shows tht for every f stisfying this property, there exist numbers > 0 nd 0 < d < suh tht Pr[S k] d k. 4 Strhler numbers nd... We hve exhusted neither the list of properties of the Strhler number, nor the works tht hve obtined them or used them. To prove the point, we mention some more ppers. In 978, Ehrenfeuht et l. introdued the sme onept for derivtion trees w.r.t. ET0L systems in [7] where it ws lled tree-rnk. Meggido et l. introdued in 98 the serh number of n undireted tree []: the miniml number of polie offiers required to pture fugitive when polie offiers my move long edges from one node to nother, nd the fugitive n move from n edge to n inident one s long s the ommon vertex is not bloked by polie offier; the fugitive is ptured when he nnot move nymore. For trees, the serh number oinides with the better known pthwidth (see e.g. []), defined for generl grphs. In order to relte the pthwidth

11 to the Strhler number, we need to extend the definition of the ltter to undireted trees: let the Strhler number S(t) of n undireted tree be the miniml Strhler number of ll the direted trees obtined by hoosing node s root, nd orienting ll edges wy from it. We n show tht for ny tree t: pthwidth(t) S(t) pthwidth(t) Currently, we re studying the Strhler number in the ontext of nturl lnguge proessing. Rell tht the Strhler number mesures the miniml height of stk required to trverse tree, or, more informlly, the miniml mount of memory required to proess it. We onjeture tht most sentenes of nturl lnguge should hve smll Strhler number simply not to overburden the reder or listener. Tble ontins the results of n exmintion of severl publily vilble tree bnks (bnks of sentenes tht hve been mnully prsed by humn linguists), whih seem to support this onjeture. For eh lnguge we hve omputed the verge nd mximum Strhler number of the prse trees in the orresponding tree bnk. We re urrently investigting whether this ft n be used to improve unlexilized prsing of nturl lnguges. Lnguge Soure Averge Mximum Bsque SPMRL. 3 English Penn.38 4 Frenh SPMRL.9 4 Germn SPMRL.94 4 Germn TueB-D/Z.3 4 Hebrew SPMRL.44 4 Hungrin SPMRL. 4 Koren SPMRL.8 4 Polish SPMRL.68 3 Swedish SPMRL.83 4 Tble. Averge nd mximum Strhler numbers for severl treebnks of nturl lnguges. : SPMRL shred tsk dtset, : 0% smple from the Penn treebnk shipped with python nltk, : TueB-D/Z treebnk. 5 Conlusions We hve skethed the history of the Strhler number, whih hs been redisovered surprising number of times, reeived surprising number of different nmes (strem order, strem rnk, index, tree rnk, tree dimension, k-terpillr... ), nd turns out to hve surprising number of pplitions nd onnetions (Prikh s theorem, Newton s method, pthwidth... ).

12 This pper is by no mens exhustive, nd we pologize in dvne to the mny uthors we hve surely forgotten. We intend to extend this pper with further referenes. If you know of further work onneted to the Strhler number, plese ontt us. 6 Aknowledgments We thnk Crlos Esprz for his help with some lultions. Referenes. D. Bienstok, N. Robertson, P. Seymour, nd R. Thoms. Quikly exluding forest. Journl of Combintoril Theory, Series B, 5():74 83, 99.. S. Bozplidis. Equtionl elements in dditive lgebrs. Theory Comput. Syst., 3(): 33, Tomás Brázdil, Jvier Esprz, Stefn Kiefer, nd Mihel Luttenberger. Speeffiient sheduling of stohstilly generted tsks. Inf. Comput., 0:87 0, Mihl Chytil nd Burkhrd Monien. Cterpillrs nd ontext-free lnguges. In Christin Choffrut nd Thoms Lenguer, editors, STACS, volume 45 of Leture Notes in Computer Siene, pges Springer, L. Devroye nd P. Kruszewski. A note on the Horton-Strhler number for rndom trees. Inf. Proess. Lett., 56():95 99, M. Droste, W. Kuih, nd H. Vogler. Hndbook of Weighted Automt. Springer, Andrzej Ehrenfeuht, Grzegorz Rozenberg, nd Dirk Vermeir. On et0l systems with finite tree-rnk. SIAM J. Comput., 0():40 58, A. P. Ershov. On progrmming of rithmeti opertions. Comm. ACM, (8):3 9, J. Esprz, S. Kiefer, nd M. Luttenberger. On fixed point equtions over ommuttive semirings. In STACS, volume 4393 of LNCS, pges Springer, J. Esprz, S. Kiefer, nd M. Luttenberger. Computing the lest fixed point of positive polynomil systems. SIAM J. Comput., 39(6):8 335, 00.. J. Esprz, S. Kiefer, nd M. Luttenberger. Newtonin progrm nlysis. J. ACM, 57(6):33, 00.. Jvier Esprz, Pierre Gnty, Stefn Kiefer, nd Mihel Luttenberger. Prikhs theorem: A simple nd diret utomton onstrution. Inf. Proess. Lett., ():64 69, Jvier Esprz, Pierre Gnty, nd Rupk Mjumdr. Prmeterized verifition of synhronous shred-memory systems. In Shrygin nd Veith [4], pges Jvier Esprz nd Mihel Luttenberger. Solving fixed-point equtions by derivtion tree nlysis. In Andre Corrdini, Brtek Klin, nd Corin Cîrste, editors, CALCO, volume 6859 of Leture Notes in Computer Siene, pges Springer, K. Etessmi nd M. Ynnkkis. Reursive mrkov hins, stohsti grmmrs, nd monotone systems of nonliner equtions. J. ACM, 56(), 009.

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