how many nonisomorphic unrooted trees and rooted trees are there with 4 vertices?
To calculate the number of nonisomorphic unrooted trees with 4 vertices, we can use Cayley's formula. Cayley's formula states that the number of unrooted trees with n vertices is equal to n^(n-2). Therefore, for 4 vertices, the number of nonisomorphic unrooted trees is 4^(4-2) = 4^2 = 16.
To calculate the number of nonisomorphic rooted trees with 4 vertices, we can use the Prüfer sequence. The Prüfer sequence is a unique sequence of n-2 integers that uniquely determines a labeled tree with n vertices.
In this case, we have n = 4 vertices, so the Prüfer sequence will have 2 elements. The possible values for these elements are from 1 to n (in this case, 1 to 4). Therefore, there are 4 * 4 = 16 different Prüfer sequences.
However, not all Prüfer sequences generate distinct rooted trees. Some sequences may correspond to isomorphic rooted trees. To exclude these duplicates, we need to count the number of ways we can arrange the 2 elements in the sequence. This can be calculated using the formula n! / (p1! * p2! * ... * pn!), where n is the total number of elements and p1, p2, ..., pn are the occurrences of each element in the sequence.
In this case, there are 2 elements in the sequence, both occurring once. So, the number of ways to arrange these elements is 2! = 2.
Therefore, the number of nonisomorphic rooted trees with 4 vertices is 16 / 2 = 8.
To recap:
Number of nonisomorphic unrooted trees with 4 vertices = 16
Number of nonisomorphic rooted trees with 4 vertices = 8