A Note on the Definition of a Tree

Very often trees appear in computer science. For example, information is often stored in treelike data structures and the execution of many recursive programs can be regarded as a traversal of a tree. Moreover, trees are efficiently used to verify many very difficult problems which are still open now. As known, leaves in a tree play an important role in the Steiner tree problem. We show that a connected graph G is a tree if and only if n₁(G) = 2+Sigma_dges3(d-2)n_d(G), where n_iG) is the number of vertices of degree i of G with respect to delta(G) les i les Delta(G), and the number phi(G) of faces of a planar graph G can be expressed as phi(G) = 2 + 1/2 Sigma_dges3(d - 2)n_d(G).