Number of labelled trees

A tree with $n$ vertices called a labelled tree if its vertices are distinguished from one another by names such as, $l_1,l_2,\ldots ,l_n$. Even if two trees are isomorphic, trees with having different vertex labels are considered as distinct graphs. According to Cayley's tree formula, there are $n^{n-2}$ labelled trees on $n$ vertices. Pr\"ufer used a simple way to prove this formula and demonstrated that there exists a bijection between the set of labelled trees on $n$ vertices and sequences of $n-2$ numbers, each in the range $0,1,2,\ldots,n-1$. Such a number sequence is called a Pr\"ufer code and it provides an alternative to the usual representation of trees. In this study, a computer algebra system (Maple) library containing various algorithms for trees is presented with the help of Pr\"ufer code. Moreover, the number of labelled trees for various families of trees such as double star, spider, centipede, firecracker, etc. is calculated using combinatorial methods with the help of Pr\"ufer code.