OFFSET
1,2
COMMENTS
The Strahler number of a full binary tree is obtained by the following process: label the external vertices (i.e., leaves) by 1 and label an internal vertex recursively as a function of the labels of its sons: if they are distinct, take the largest of the two and if they are equal, increase them by 1. The Strahler number is the label of the root.
LINKS
P. Flajolet, J.-C. Raoult, and J. Vuillemin, The number of registers required for evaluating arithmetic expressions, Theoret. Comput. Sci. 9 (1979), no. 1, 99-125.
J. Francon, Sur le nombre de registres nécessaires à l'évaluation d'une expression arithmétique, RAIRO Inform. Theor, 18, 1984, 355-364.
Xavier Gérard Viennot, A Strahler bijection between Dyck paths and planar trees, FPSAC (Barcelona, 1999), Discrete Math. 246 (2002), no. 1-3, 317-329. MR1887493 (2003b:05013).
D. Zeilberger, A bijection from ordered trees to binary trees that sends the pruning order to the Strahler number, Discrete Math., 82, 1990, 89-92.
FORMULA
a(n) = Sum_{k>=1} k*A127151(n,k).
G.f.: Sum_{k>=1} k*F[k], where F[k] = F[k](z) (k=1,2,...) are defined recursively by F[k] = zF[k-1]^2 + 2zF[k](F[0] + F[1] + ... + F[k-1]), with F[0]=1.
MAPLE
F[0]:=1: for k from 1 to 4 do F[k]:=simplify(z*F[k-1]^2/(1-2*z*sum(F[j], j=0..k-1))) od: g:=add(j*F[j], j=1..4): gser:=series(g, z=0, 32): seq(coeff(gser, z, n), n=1..28);
MATHEMATICA
f[0]=1; For[k=1, k <= 4, k++, f[k] = Simplify[z*f[k-1]^2/(1-2*z*Sum[f[j], {j, 0, k-1}])]]; g=Sum[j*f[j], {j, 1, 4}]; gser=Series[g, {z, 0, 25}]; CoefficientList[gser, z] // Rest (* Jean-François Alcover, Nov 22 2012, translated from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jan 06 2007
STATUS
approved