R. Otter, The number of trees, Ann. of Math, 49, 1948, 583-599.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....size and order n. However a complete determination of (n) in the general case is deferred to a later paper. 2 Exact Counting As is usual for tree counting, rooted trees are considered first. Then unrooted (free) trees are counted with the help of Otter s dissimilarity characteristic equation [10]. Let R(n) denote the number of identity rooted oriented trees of order n, and let R(x) n1 R(n)x be the ordinary generating function. There will be no occasion to substitute an integer value for the variable x, so this notation should cause no confusion. In R. Simion s paper [11] R(n) ....

R. Otter, The number of trees, Ann. of Math. 49 (1948) 583-599.

....and t n , and the generating functions by and t(x) t n x The structure of these trees is much more dicult than that of rooted trees, where the successors have a left to right order. It turns out that one has to apply P olya s theory of counting and an amazing observation (8) by Otter [15]. Theorem 6. The generating functions t(x) and t(x) satisfy the functional equations t(x) x exp t(x) 1 1 (7) t(x) 1 ) 8) They have a common radius of convergence 0:338219 which is given by t( 1, that is, t(x) is convergent at x = Furthermore, they ....

....given by t n = 11) t n = 12) Remark . In 1937 P olya [16] already discussed the generating function t(x) and showed that the radius of convergence satis es 0 1 and that x = is the only singularity on the circle of convergence jxj = Later Otter [15] showed that t( 1 and used the representation (9) to deduce the asymptotics for t n . He also calculated 0:338219 and b 2:6811266. However, his main contribution was to show (8) Consequently he derived (10) and (12) Proof. We rst show (7) As in the previous cases we can think of ....

R. Otter, The number of trees, Ann. Math. 49 (1948), 583-599.

....components (trees) Initially, one could apply the de nitions indicated in Section 2.3 to observe that the set F of all unlabelled (i.e. isomorphism classes of) nite forests forms an additive arithmetical semigroup, whose prime elements are the unlabelled trees. A famous theorem of R. Otter [15] states that the total number T (n) of unlabelled trees with n vertices satis es (n) C 0 q where C 0 and q 0 are explicitly described positive constants (q 0 1) The proof is based partly on some analytic methods of P olya [17] Many decades after the original contributions of ....

....P olya and Otter, E.M. Palmer and A.J. Schwenk [16] used both P olya s methods and Otter s theorem to estimate the corresponding total number F (n) of all unlabelled forests with n vertices. They showed that (n) K 0 C 0 q where K 0 1 is also an explicitly described constant. Otter [15] also proved a variation of his theorem above for the total number r (n) of unlabelled rooted trees with n vertices: r (n) C 0 q Since it is well known that the number F r (n) of unlabelled rooted forests with n vertices equals T r (n 1) it follows (as Palmer and Schwenk ....

R. Otter, The number of trees. Ann. of Math., 49 (1948), 583-599.

....t n = jT n j and t (r) n = jT (r) n j. In 1937 P olya [7] already discussed the generating function t (r) x) X n1 t (r) n x n and showed that the radius of convergence ae satisfies 0 ae 1 and that x = ae is the only singularity on the circle of convergence jxj = ae. Later Otter [6] showed that t (r) ae) 1 and used the asymptotic expansion t (r) x) 1 Gamma b(ae Gamma x) 1=2 c(ae Gamma x) d(ae Gamma x) 3=2 Delta Delta Delta (1.1) to deduce that t (r) n b p ae 2 p n Gamma3=2 ae Gamman : Note that c = b 2 =3 2:3961466. He also ....

R. Otter, The number of trees, Ann. Math. 49 (1948), 583--599.

....the generating function is given, its coecients can be computed by dynamic programming. Harary and Palmer s book [23] contains a survey on using generating functions to do unlabelled enumeration. Their book gives a full treatment of the enumeration of unlabelled trees, following the work of Otter [43]. In order to illustrate the principles, we repeat a few of the details here. Let T (x) P 1 n=1 T n x n be the generating function for rooted unlabelled trees. That is, T n is the number of rooted unlabelled trees with n vertices. P olya s theorem gives an expression 4 for T (x) which can ....

R. Otter, The number of trees, Ann. of Math. 49 (1948), 583-599.

....for almost all graphs. It turns out that almost all graphs are of diameter 2. 2 Since many inputs are in the form of trees, either rooted or free, one might ask about probable behavior of trees. When people started studying the probabilistic behavior of trees in the 1950s and 1960s (e.g. [O48], Cl58] Re60] etc. they followed in the tradition of Cayley [Ca81] who had used generating functions in [Ca89] see [Pal85] Indeed, generating functions seem to be the most popular tool for this sort of thing. Definition 1 Fix a sequence Omega = Omega 0 ; Omega 1 ; of sets of ....

R. Otter, The Number of Trees, Ann. of Math. 49 (1948), 583--599.

....In the unlabelled case, it is easy to handle rooted trees, since the number of forests of rooted trees on n vertices is equal to the number of rooted trees on n 1 vertices. Take a new root, and join it to all the old roots. Since these numbers grow exponentially with constant 2.95576. [24], the limiting probability of connectedness is the reciprocal of this number, namely 0.33832. It appears that exponential growth for the number of n element unlabelled structures is necessary for the probability of connectedness to be strictly between 0 and 1, though I cannot prove such a ....

....the hexagon two graph as an induced substructure. Moreover, non isomorphic trees give rise to non isomorphic two graphs. This solves the counting problem for unlabelled pentagon and hexagon free two graphs: the number on n points is equal to the number of trees with n edges, calculated by Otter [24]. 25 However, there is a further difficulty associated with counting the labelled pentagon and hexagon free two graphs. For example, a path with n edges can have its edges labelled in n =2 different ways, but all of these give rise to the null two graph (the two graph with no triples) The ....

R. Otter, The number of trees, Ann. Math. (2) 49 (1948), 583--599.

....where the generating functions typically have only algebraic singularities. To oversimplify a little, it can be said that at least asymptotic enumeration of unlabeled trees of various kinds is very well understood. The basic method used there was developed by Polya [32] and perfected by Otter [31]. Their method is presented also in [16] and it is so well understood that a few years ago a paper was written with the title Twenty step algorithm for determining the asymptotic number of trees of various species [17] The word algorithm in the title of that paper should not be interpreted ....

R. Otter, The number of trees, Ann. Math. 49 (1948), 583-599.

.... p 1 f ffi g p 1 = i p 1 f p 1 j ffi g p 1 = i p 1 g p 1 j ffi f p 1 ffi g p 1 : Cancellation proves that g f ffi g p 1 = p 1 g p 1 ; and hence that f = Lg. For example, Lh 2 = e 2 and vice versa. The following theorem is related to results of Otter [29] and Hanlon Robinson [15] on the enumeration of unrooted trees. 7.17) Theorem. Let V be a cyclic S module such that V( n) 0 for n 2 and V( n) is finite dimensional for all n. Define the elements of f = e 2 Gamma Ch(V) and g = h 2 Ch(TV) Then g = Lf . Proof. By definition of L, we ....

R. Otter, The number of trees, Ann. Math. 49 (1948), 583--599.

....interpretation of Construction 2 is known. To conclude this section, I consider briefly the enumeration problems. According to Proposition 2, the number of unlabelled 5,6 free two graphs is equal to the number of trees with n edges (that is, with n 1 vertices) This number was found by Otter [6]; I will not reproduce the formula here. The sequence is listed as number 299 in Sloane [11] Cayley s famous theorem [4] shows that the number of labelled trees on n vertices is n n Gamma2 . It follows that the number of trees with n labelled edges is (n 1) n Gamma2 for n 2 (this will be ....

R. Otter, The number of trees, Ann. Math. (2) 49 (1948), 583--599.

....paper. As we expect the method will be unfamiliar to many readers, and as many of the papers concerned contain much very interesting, but for the present purposes distracting, additional material, we will briefly sketch the method. Further examples of its use, and justification, can be found in [18, 3, 8, 9, 6, 16, 30, 27, 31], as well as the already cited early paper of Riordan and Shannon In the applications of interest, the number of trees of a certain sort with n leaves will be determined by the (real, nonnegative) coefficient of x n in an exponential or ordinary generating series T (x) T 1 x T 2 x 2 ....

....are exactly L leaves, each having one of 2k labels X 1 ; X 1 ; X 2 ; X 2 ; X k ; X k . Consider the corresponding ordinary generating series T k (x) 1 X L=1 T kL x L : By a standard argument (e.g. using Polya s Enumeration Theorem applied to the symmetric group Sm see e.g. [8, 18, 20, 21, 22, 27]; or directly see e.g. 16, 23] the ordinary generating series for a forest of m such trees is X j 1 ; j m j 1 2j 2 Delta Delta Delta mj m=m (T k (x) j 1 j 1 (T k (x 2 ) j 2 2 j 2 j 2 Delta Delta Delta (T k (x m ) jm m jm j m : As each tree consists of ....

R. Otter. The number of trees. Ann. of Math., 49:583--599, 1948.

....0:3367 : for the unlabelled structures. The latter holds because there is a natural bijection between forests of rooted trees on n vertices, and rooted trees on n 1 vertices; and 2:997 : is the exponential constant in the asymptotic formula for the number of unlabelled trees: see Otter [11]. For the labelled case, see R enyi [15] There is no need to restrict ourselves to graphs. The probability of connectedness of a random N free poset tends to ( p 5 Gamma 1) 2, in both the labelled and the unlabelled case (El Zahar [4] see also Stanley [18] Incidentally, there is no simple ....

R. Otter, The number of trees, Ann. Math. (2) 49 (1948), 583--599.

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R. Otter, The number of trees, Ann. of Math, 49, 1948, 583-599.

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R. Otter, The Number of Trees, Ann. of Math. 49 (1948), 583-- 599.

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R. Otter, The number of trees, Ann. of Math, 49, 1948, 583--599.

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R. Otter, The number of trees, Ann. of Math., 49 (1948), pp. 583--599.

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R. Otter, The Number of Trees, Ann. of Math. 49 (1948), 583-- 599.

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R. OTTER, The number of trees, Ann. of Math. 49 (1948), 583-599.

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