

A214574


The Strahler number of the rooted tree with MatulaGoebel number n.


1



1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3
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OFFSET

1,4


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; (ii) 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 MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.


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 MatulaGoebel 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 tth 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 Strahler number a(n) = degree(G(n)).


EXAMPLE

a((4)=2 because the rooted tree with MatulaGoebel number 4 is V; the two leaves have Strahler numbers 1,1, and the root has Strahler number 2; this is  by definition  the Strahler number of the tree.


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: seq(degree(G(n)), n = 1 .. 200);


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



