OFFSET
1,2
COMMENTS
The Strahler number of a vertex of a rooted tree is defined recursively in the following way: (i) the Strahler number of a leaf is 1; if the vertex has one child with Strahler number i and all other children have Strahler number less than i, then the Strahler number of the vertex is again i; (iii) if the vertex has two or more children with Strahler number i and no child with Strahler number greater than i, then the Strahler number of the vertex is i+1. See the Wikipedia reference.
The Strahler number of a rooted tree T is defined as the Strahler number of the root of T.
The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
Number of entries in row n = A214574(n) = the Strahler number of the rooted tree with Matula-Goebel number n.
Sum of entries in row n = A061775(n) = number of vertices in the rooted tree with Matula-Goebel number n.
LINKS
Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
Emeric Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
F. Göbel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Wikipedia, Strahler number
FORMULA
Define the Strahler polynomial of a rooted tree T as the generating polynomial of the vertices of T with respect to their Strahler numbers. For example, it follows at once that the Strahler polynomial of the rooted tree V is 2x + x^2. Denote by G(n)=G(n;x) the Strahler polynomial of the rooted tree with Matula-Goebel number n. Clearly, A214573(n,k) is the coefficient of x^k in G(n). We have (i) G(1)= x; (ii) if n=p(t) (the t-th prime), then G(n) = x^{degree(G(t)} + G(t); (iii) if n=rs (r,s>=2), then G(n) = G(r) - degree (G(r)) + G(s) - degree(G(s)) + x^m, where m = 1+degree(G(r)) if degree(G(r))=degree(G(s)) and m = max(degree(G(r), G(s)) otherwise. The Maple program is based on these recursion relations.
EXAMPLE
Row 4 is 2,1. Indeed, the rooted tree with Matula-Goebel number 4 is V; the two leaves have Strahler numbers 1,1, and the root has Strahler number 2.
Triangle starts:
1;
2;
3;
2,1;
4;
3,1;
2,2;
...
MAPLE
with(numtheory): G := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then x elif bigomega(n) = 1 then sort(expand(x^degree(G(pi(n)))+G(pi(n)))) elif 1 < bigomega(n) and degree(G(r(n))) <> degree(G(s(n))) then sort(G(r(n))-x^degree(G(r(n)))+G(s(n))-x^degree(G(s(n)))+x^max(degree(G(r(n))), degree(G(s(n))))) else sort(G(r(n))-x^degree(G(r(n)))+G(s(n))-x^degree(G(s(n)))+x^(1+degree(G(r(n))))) end if end proc: for n to 100 do g[n] := G(n) end do: for n to 100 do seq(coeff(g[n], x, j), j = 1 .. degree(g[n])) end do: seq(seq(coeff(g[n], x, j), j = 1 .. degree(g[n])), n = 1 .. 100);
MATHEMATICA
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
degree = Exponent[#, x]&;
G[n_] := G[n] = Which[n == 1, x, PrimeOmega[n] == 1, Sort[ Expand[ x^degree[G[PrimePi[n]]] + G[PrimePi[n]]]], 1 < PrimeOmega[n] && degree[G[r[n]], x] != degree[G[s[n]]], Sort[G[r[n]] - x^degree[G[r[n]]] + G[s[n]] - x^degree[G[s[n]]] + x^Max[degree[G[r[n]]], degree[G[s[n]]]]], True, Sort[G[r[n]] - x^degree[G[r[n]]] + G[s[n]] - x^degree[G[s[n]]] + x^(1 + degree[G[r[n]]])]];
Table[Table[Coefficient[G[n], x, j], {j, 1, degree[G[n], x]}], {n, 1, 100}] // Flatten (* Jean-François Alcover, Aug 11 2024, after Emeric Deutsch *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Aug 14 2012
STATUS
approved