|
| |
|
|
A214573
|
|
Irregular triangle read by rows: the (n,k)-entry is the number of vertices with Strahler number k in the rooted tree with Matula-Goebel number n (n>=1, k>=1).
|
|
3
|
|
|
|
1, 2, 3, 2, 1, 4, 3, 1, 2, 2, 3, 1, 4, 1, 4, 1, 5, 4, 1, 3, 2, 3, 2, 5, 1, 4, 1, 2, 3, 5, 1, 3, 2, 5, 1, 4, 2, 5, 1, 4, 2, 5, 1, 6, 1, 4, 2, 6, 1, 4, 2, 4, 2, 6, 1, 6, 5, 1, 6, 1, 3, 3, 5, 2, 6, 1, 4, 2, 4, 2, 5, 2, 6, 1, 3, 3, 5, 2, 3, 3, 6, 1, 7, 1, 5, 2, 5, 2, 6, 1, 4, 2, 1, 7, 1, 4, 3, 5, 2, 4, 2, 7, 1, 7, 1, 5, 2, 5, 2, 5, 2, 2, 4, 7, 1, 5, 2, 6, 1, 6, 2, 6, 1, 6, 2, 7, 1, 3, 3, 4, 3, 6, 2, 6, 2, 5, 2, 7, 1, 4, 3, 5, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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.
|
|
|
REFERENCES
|
E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
F. Goebel, 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 Y-N. 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 Review, 10, 1968, 273.
|
|
|
LINKS
|
|
|
|
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);
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,tabf
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
