Cayley Tree

A Cayley tree is a tree in which each non-leaf graph vertex has a constant number of branches  is called an -Cayley tree. 2-Cayley trees are path graphs. The unique -Cayley tree on  nodes is the star graph. The illustration above shows the first few 3-Cayley trees (also called trivalent trees, binary trees, or boron trees). The numbers of binary trees on , 2, ... nodes (i.e., -node trees having vertex degree either 1 or 3; also called 3-Cayley trees, 3-valent trees, or boron trees) are 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0 ,4, 0, 6, 0, 11, ... (OEIS A052120).

The illustrations above show the first few 4-Cayley and 5-Cayley trees.

The percolation threshold for a Cayley tree having  branches is

REFERENCES

Sloane, N. J. A. Sequence A052120 in "The On-Line Encyclopedia of Integer Sequences."