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A358724 Difference between the number of internal (non-leaf) nodes and the edge-height of the rooted tree with Matula-Goebel number n.

%I A358724 #5 Dec 01 2022 08:55:36
%S A358724 0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,1,0,2,0,2,0,0,1,0,0,1,0,
%T A358724 1,1,0,0,1,0,0,1,0,0,2,1,1,0,1,2,1,0,0,2,2,0,1,0,0,1,1,0,2,0,2,1,0,0,
%U A358724 2,1,0,1,1,0,3,0,1,1,0,0,3,0,1,1,2,0,1
%N A358724 Difference between the number of internal (non-leaf) nodes and the edge-height of the rooted tree with Matula-Goebel number n.
%C A358724 Edge-height (A109082) is the number of edges in the longest path from root to leaf.
%C A358724 The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
%F A358724 a(n) = A342507(n) - A109082(n).
%e A358724 The tree (o(o)((o))(oo)) with Matula-Goebel number 210 has edge-height 3 and 5 internal nodes, so a(210) = 2.
%t A358724 MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A358724 Table[Count[MGTree[n],_[__],{0,Infinity}]-(Depth[MGTree[n]]-2),{n,100}]
%Y A358724 Positions of 0's are A209638, complement A358725.
%Y A358724 Positions of 1's are A358576, counted by A358587.
%Y A358724 Other differences: A358580, A358726, A358729.
%Y A358724 A000081 counts rooted trees, ordered A000108.
%Y A358724 A034781 counts rooted trees by nodes and height, ordered A080936.
%Y A358724 A055277 counts rooted trees by nodes and leaves, ordered A001263.
%Y A358724 MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
%Y A358724 MG core: A000040, A000720, A001222, A007097, A056239, A112798.
%Y A358724 Cf. A185650, A206487, A316321, A358577, A358578, A358592, A358730.
%K A358724 nonn
%O A358724 1,25
%A A358724 _Gus Wiseman_, Nov 29 2022