Conditions for Spanning Trees Whose Internal Subtrees Have Few Branch Vertices and Leaves

Let T be a tree. The sets of leaves and branch vertices of T are denoted by L(T) and B(T), respectively. For two distinct vertices u, v of T, let PT[u, v] denote the unique path in T connecting u and v. When B(T) &NOTEQUexpressionL; 0, we call the graph ST = boolean OR(u,vEB(T)) PT [u, v] the internal subtree of T. In this paper, we give two conditions for a connected graph to have a spanning tree whose internal subtree has few branch vertices and leaves. Moreover, the sharpness of our result is also shown.