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Cayley Tree


CayleyTrees

A Cayley tree is a tree in which each non-leaf graph vertex has a constant number of branches n is called an n-Cayley tree. 2-Cayley trees are path graphs. The unique n-Cayley tree on n+1 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 n=1, 2, ... nodes (i.e., n-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).

Cayley4Cayley5

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

The percolation threshold for a Cayley tree having z branches is

 p_c=1/(z-1).

See also

Cayley Graph, Path Graph, Star Graph, Tree

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References

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

Cite this as:

Weisstein, Eric W. "Cayley Tree." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CayleyTree.html

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