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A tree with 5 vertices has 4 edges. So, the total degree of this graph is 8.
We need to find all degree sequences such that the total degree is 8 and each vertex has degree at least 1.
Let us consider the maximum degree of the graph. Since the graph has 4 edges, it follows that maximum degree . Since the total degree is 8, it follows that maximum degree .
So, maximum degree .
If the maximum degree is 4, then we have the following degree sequnce: (4, 1, 1, 1, 1).
If the maximum degree is 3, then we have: (3, 2, 1, 1, 1).
If the maximum degree is 2, then we have: (2, 2, 2, 1, 1).
There are three unlabelled trees with 5 vertices.