*N.B. A detailed on-line essay by S. Finch
was the starting point for this entry.*

A Rooted Tree for which the Root is adjacent to at most two Vertices, and all nonroot Vertices are adjacent to at most three Vertices. Let be the number of weakly binary trees of order , then . Let

(1) |

(2) | |||

(3) | |||

(4) | |||

(5) |

Otter (Otter 1948, Harary and Palmer 1973, Knuth 1969) showed that

(6) |

(7) |

(8) |

(9) |

(10) |

(11) | |||

(12) |

giving

(13) |

**References**

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

Harary, F. *Graph Theory*. Reading, MA: Addison-Wesley, 1969.

Harary, F. and Palmer, E. M. *Graphical Enumeration.* New York: Academic Press, 1973.

Knuth, D. E. *The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 2nd ed.* Reading, MA: Addison-Wesley, 1973.

Otter, R. ``The Number of Trees.'' *Ann. Math.* **49**, 583-599, 1948.

© 1996-9

1999-05-26