A064060 Number of connected, homeomorphically irreducible (also called series-reduced) trees with n >= 2 labeled leaves (numbers in nondecreasing order).

1, 1, 1, 3, 1, 10, 15, 1, 10, 15, 15, 45, 60, 90, 1, 21, 35, 70, 105, 105, 105, 105, 210, 315, 420, 630, 630, 1, 28, 35, 56, 105, 168, 210, 210, 280, 280, 280, 315, 420, 420, 560, 560, 840, 840, 840, 1260, 1260

Offset: 2

Author

Wolfdieter Lang and Christoph Mayer (Christoph.Mayer(AT)dlr.de), Sep 13 2001

Keywords

• nonn
• tabf

Comments

The number of entries of row n of this array is A007827(n), n >= 2.

With v the total number of nodes (vertices), e the number of edges (links), n >= 2 the number of edges ending in a degree 1 node (leaves), i the number of edges which end in nodes with degree >=3 (internal edges) and v_{d} the number of nodes of degree d=1,3,4,... one has: v = e+1 = n + Sum_{d>=3}v_{d}, i = e-n, Sum_{d>=3}d*v_{d} = 2(v-1)-n.

Examples

			Irregular array starts:
{1};
{1};
{1, 3};
{1, 10, 15};
{1, 10, 15, 15, 15, 45, 60, 90};
{1, 21, 35, 70, 105, 105, 105, 105, 210, 315, 420, 630, 630};
{1, 28, 35, 56, 105, 168, 210, 210, 280, 280, 280, 315, 420, 420, 560, 560, 840, 840, 840, 1260, 1260, 1680, 1680, 1680, 1680, 2520, 2520, 2520, 2520, 3360, 5040, 5040};
...
		

Links

• Ch. Mayer, Illustration
• Index entries for sequences related to trees

Crossrefs

The row sums give A000311(n-1), n >= 2. Cf. A007827.