I predicted the formula by finding total number of non-isomorphic labelled trees of order 1 is 1 , order 2 is 1,order 3 is 3,order 4 is 16,order 5 is 125.But how do i prove it ? I am beginner in graph theory so i will be very thankful if i get some simple approach to solve the problem(Not by using some difficult theorems).
2026-03-25 17:42:08.1774460528
How to prove that total number of non-isomorphic labelled trees of order $n$ is $n^{n-2}$?
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One approach is to use Prüfer sequences; there is a bijection between labelled trees and length-$n-2$ sequences of numbers in $[1,n]$ (the Wikipedia article gives a good description of how to go from sequence to tree and back). Since there are $n^{n-2}$ such sequences, there are also this many labelled trees.