## Rooted Tree

A Tree with a special node called the Root'' or Eve.'' Denote the number of rooted trees with nodes by , then the Generating Function is
 (1)
(Sloane's A000081). This Power Series satisfies

 (2) (3)

where is the Generating Function for unrooted Trees. A Generating Function for can be written using a product involving the sequence itself as
 (4)

The number of rooted trees can also be calculated from the Recurrence Relation

 (5)

with and , where the second sum is over all which Divide (Finch).

References

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/otter/otter.html

Ruskey, F. Information on Rooted Trees.'' http://sue.csc.uvic.ca/~cos/inf/tree/RootedTree.html.

Sloane, N. J. A. Sequence A000081/M1180 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.