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

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, 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, 2, 1, 0, 1, 1, 0, 3, 0, 1, 1, 0, 0, 3, 0, 1, 1, 2, 0, 1

Offset

1,25

Comments

Edge-height (A109082) is the number of edges in the longest path from root to leaf.

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.

Links

Table of n, a(n) for n=1..87.

Formula

a(n) = A342507(n) - A109082(n).

Example

The tree (o(o)((o))(oo)) with Matula-Goebel number 210 has edge-height 3 and 5 internal nodes, so a(210) = 2.

Mathematica

MGTree[n_]:=If[n==1, {}, MGTree/@Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Count[MGTree[n], _[__], {0, Infinity}]-(Depth[MGTree[n]]-2), {n, 100}]

Crossrefs

Positions of 0's are A209638, complement A358725.

Positions of 1's are A358576, counted by A358587.

Other differences: A358580, A358726, A358729.

A000081 counts rooted trees, ordered A000108.

A034781 counts rooted trees by nodes and height, ordered A080936.

A055277 counts rooted trees by nodes and leaves, ordered A001263.

MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.

MG core: A000040, A000720, A001222, A007097, A056239, A112798.

Cf. A185650, A206487, A316321, A358577, A358578, A358592, A358730.

Sequence in context: A269248 A092078 A360071 * A325135 A263577 A343631

Adjacent sequences: A358721 A358722 A358723 * A358725 A358726 A358727

Keyword

nonn

Author

Gus Wiseman, Nov 29 2022

Status

approved