A214574 The Strahler number of the rooted tree with Matula-Goebel number n.
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
Offset: 1
Keywords
Examples
a(4)=2 because 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; this is - by definition - the Strahler number of the tree.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..65537
- 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. 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 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
- Index entries for sequences related to Matula-Goebel numbers
Crossrefs
Programs
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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);
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PARI
A214574(n) = if(1==n, 1, if(isprime(n), A214574(primepi(n)), my(f=factor(n), m, mpi=0); f[,1]=apply(A214574,f[,1]); m = vecmax(f[,1]); for(i=1,#f~,if(m==f[i,1], if(mpi || f[i,2]>1, return(1+m), mpi = i))); (m))); \\ Antti Karttunen, Jan 20 2025
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 Strahler number a(n) = degree(G(n)).
Comments