# Thread: Caley's formula

1. ## Caley's formula

Hope someone can help with the above please? There are three non-isomorphic trees available on 5 vertices as shown in the graphic. The first tree in the picture can be labelled in 5 ways - which I can see using (n!/(k!(n-k)!)

I am told the second and third can be labelled in 60 ways. Could someone explain how the figure of 60 is arrived at?

Many Thnaks

Blobfish

2. ## Re: Caley's formula

For #2

$5 \text{ ways to select the level 0 vertice. Then }\dbinom{4}{3}\text{ ways to select the 3 level 1 vertices. Then }\dbinom{3}{1}\text{ ways to select which vertice takes the level 2 child.}$

$5 \cdot 4 \cdot 3 = 60$

For #3

$5 \text{ ways to select the level 0 vertice. Then }\dbinom{4}{2}\text{ ways to select the 2 level 1 vertices. Then }\dbinom{2}{1}\text{ ways to arrange the level 2 children.}$

$5 \cdot 6 \cdot 2 = 60$