On the Distribution o f Vertex-Degrees in a Strata o f a Random Recursive Tree. Marian D O N D A J E W S K I and Jerzy S Z Y M A N S K I

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1 BULLETIN DE L ACADÉMIE POLONAISE DES SCIENCES Série des sciences mathématiques Vol. XXX, No. 5-6, 982 COMBINATORICS On the Distribution o f Vertex-Degrees in a Strata o f a Rom Recursive Tree by Marian D O N D A J E W S K I Jerzy S Z Y M A N S K I Presented by C R Y L L -N A R D Z E W S K I on September 4, 98 Summary. We shall derive a formula for the expected value Ar {n,k ) of the number of vertices of degree r in Sk of a rom recursive tree with n vertices. A strata Sk is a set o f such vertices of a tree that their distance from the root is equal to k.. Introduction. A tree Rn with n vertices labelled,2,..., n is a recursive tree if n = or if n ^ 2 the и-th vertex is joined to one o f n vertices o f R _ j. A vertex with label is a root o f a recursive tree (see [ ] for definitions not given here). A strata Sk is a set o f such vertices o f a tree that their distance from root is equal to k, where k = 0,,..., n. Let Ar (и, к) denote the expected value o f the number o f vertices o f degree r in Sk over the family o f all recursive trees with n vertices. Moreover, let у4г (n) denote the expected value of the number o f vertices o f degree r in a tree R (the root is not taken into account). Our main object here is to determine Ar(n,k). Summing Ar(n,k) over к we can calculate Ar (и) compare it with the results obtained by Na Rapoport [7]. 2. Results for end-vertices. Let ц (n, k) denote the expectation o f the number o f vertices in Sk. Meir M oon [5 ] have shown that ц(п,к) fulfills the following recursive equation ( ) li (и +, к + ) = pi (n, к -f ) + Ц (n, k)/n, where ц(п, 0) = ц (n, k) = 0 for k^ n. They also proved that - к! ^ flq û 2 a* where the sum is over all positive integers ah such that ^ a, ^ n cti Ф üj for ^ i, j ^ к i Ф j. [205]

2 206 M. Dondajewski, J. Szymanski W e have found that gin,k) can be presented in the following form T heorem. s (и, k+ ) (2) /t(n,k) = (- l)" +k+ (л-)! where s(n,k) denotes the Stirling numbers of the first kind. P r o o f. It is easy to see that we can rewrite equation () as follows (3) b (n +, k) = b in, k i) n b (n, k), where b (in, k) = ( - l)n+* + ( n - )! g (n, k-\). Obviously (3) is the recursive equation for the Stirling numbers of the first kind. Boundary condition b(l, ) = is also fulfilled. Using the above Theorem one can prove the following result. T heorem 2. (4) A' (n k) = - ^ T = Z - -( J T Î )! for n ^ 2, к =,2,..., n. P r o o f. It follows from the construction o f recursive trees that (5) Ax(n+, k) = Ax(n, k)-\ /j. (n, к ). n n Let f (n) {n ) Ax(n, k), then it follows from (5) that where Af (n) = f (n + l ) f (n). we get Af{n) = g {п, к - ), N ow from the well-known facts o f the theory of recursive equations (6) / (n) = c + x n a,k - )= c + У n (/, * - и, i = because g in, k) = 0 for к > n. Obviously Ax(n +, n) = fn\. The abo\e identity combined with (6) implies C = 0. This completes the proof. Summing Axin,k) over к we can prove i = к Corollary. (7) A j in) = n/2. P r o o f. л,(")= I («Д) = Ц- I f II, S (- fs(i,k) = к = i = l l* lc= l

3 Distribution o f Vertex-Degrees 207 / n- I! = ( - У 0 - )! ( )' il = n/2. So, we arrive at the thesis. Similar result was obtained by N a Rapoport [7 ] in another way. Th ey count the root additionally prove that the number of end-vertices in Rn is (n/2 + /(л )). 3. Vertices o f degree r ^ 2. N ow let us consider the more complicated case when r ^ 2. The following lemma is an immediate consequence o f the way in which Rn+ i is obtained from Rn. (8) L e m m a. I f r ^ 2 then Ar (n, к) = n Z /- ; = A r _ i (i, /с), (9) fo r n ^ k + r. i = k + r - Л'(") = 7~Г Z ^-i (*) C o r o l l a r y 2. I f r ^ 2 n ^ k + r then n Ar(n+\, k) = (n-\) Ar (n, k) + Ar_l (n, k), (/-) Ar (n) = Ar-i (/ l) + (/i 2) Ar (/ i- l). N ow we shall prove the following Theorem. T h e o r e m 3. If r ^ 2 n ^ k + r then ( 0) where (- )* v ( - iys(j,k) Ar {n,k) = / - k О - D! Л Г /I г л г яаг= Z Z - Z Hj,r, ц-м а-it «r-i =.,-2 * (J2+ ) (Ir - i + r _ 2) Relation (0) can easily be proved by induction by r using formula (8). C o r o l l a r y 3. If r ^ 2 n > r then () Ar (n) = 2~r n+ s ( n -, k) ( 2). P r o o f. T o prove () let us introduce a generating function: (2) G ( x, y ) = X Z A r ( n ) x r yn~ l. n= 2 r =

4 208 M. Dondajewski, J. Szymanski It converges for \x\ ^, y <. Obviously (3) G (0, y) = G (x, 0) = 0, n - G ( l, y ) = X / X А г ( п ) = X (и - l ) / = y/(l yÿ n = 2 r = n= 2 From (2), (9) (5) we get But G(x,y) = Z X Ar(n)xr yn~ l + x X A i ( и ) / _ = n = 3 г = 2 n 2 XV (2 vi * n~ -, ~ (l-y)-2+ I I I Л-(0*г/->- ). Z n = 3 r= 2 i= r 5 G fx v) 0 0 n -l n -l ', w = х ( - у ) - з + П n=3 r = 2 i= r finally dg (x,y ) dy = x (i-y )_3+ x X Л-(0*г X / ~ 2* i= 2 г = n= I + = x ( l - y ) 3 + xg (x, y)/(l y). The solution of this equation with boundary condition (3) has the form (4) But G{x,y) = ~----- (( y) 2 ( y) x). 2 x - ^ - = - l + (l- x / 2 ) - = X (*/2)*, i x k= i (-уг 2-( -у Г х = X {fc+l-(*)*/*!}/, * = П where (x)k= x (x + l)...(x + n ) = ( )" X s (и, к) ( l)k xk. lt = о Formula (4) can be rewritten as follows (5) G (x, y) = X (*/2)* X { + --!!, X s f a - U r H - l / x ' } / -. * = i я= 2 i (и )! r = i J Formula () can be obtained by comparing coefficients o f power series (2) (5).

5 Distribution o f Vertex-Degrees 209 Using the asymptotic formula of Jordan for the Stirling number o f the first kind we can easily find that for fixed r n -* оо AT(n) = n 2-r + o (). This result is the same as that of relation (49) by N a Rapoport [7]. Using relation (2) we can present (0) () in equivalent form (6) (7) Ar{n,k) = -Ц - д О 'Д - l )Hj,r, n i j = k Ar(n) = 2 г «- - ^ з у - Z М и - Д ~ ) 2 к. INSTITUTE OF MATHEMATICS, TECHNICAL UNIVERSITY, PIOTROWO 3a, POZNAN (INSTYTUT MATEMATYKI, POLITECHNIKA POZNANSKA) REFERENCES [ ] F. H arary, E. P alm er, Graphical enumeration, Academic Press, New York, 973. [2 ] J. L. G a stw irth, A probability model of a pyramid scheme, Amer. Statistician, 3 (977), [3 ] M. K aron sk i, A review o f rom graphs, Komunikaty i Rozprawy IM UAM, Poznan, 978. [4 ] A M eir, J W. M o o n. The distance between points in rom trees, J. Comb. Theory, 8 (970), [5 ] A M eir, J W. M o o n, On the altitude o f nodes in rom trees, Can. J. Math., 30 (978), [ 6] A. M eir, J. W. M o o n, Path edge-covering constants for certain families o f trees, Util. Math., 4 (978), [7 ] H. S. N a, A R a p o p o r t, Distribution o f nodes o f a tree by degree. Mathematical Biosciences, 6 (970), M. Дондаевский, E. Шиманьский, О распределении степеней вершии в слоях случайных рекуррентных деревьев Слой S/< корневого дерева это множество всех вершин, расстояние от корня которых равняется к. Пусть Аг (п, к) есть математическое ожидание числа вершин степени г в слое Sk для семьи всех рекуррентных деревьев, имеющих п вершин. Целью настоящей статьи является обозначение Аг (п, к).