Let T be a tree, a vertex of degree one is a leaf of T and a vertex of degree at least three is a branch vertex of T . The set of leaves of T is denoted by L(T ) and the set of branch vertices of T is denoted by B(T ). For two distinct vertices u, v of T , let P T [u, v] denote the unique path in T connecting u and v. Let T be a tree with B(TThe resulting graph is a subtree of T and is denoted by R Stem(T ). It is called the reducible stem of T . A leaf of R Stem(T ) is called a peripheral branch vertex of T . In this paper, we give some sharp sufficient conditions on the independence number and the degree sum for a graph G to have a spanning tree with few peripheral branch vertices.

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