For an integer k with , a tree with maximum degree at most k is called a k-tree, and a tree with at most k leaves is called a k-ended tree.
As a generalization of the above two concepts, the concept of total excess is defined as follows: For a graph T and an integer k, the total excess te(T, k) from k is defined as

te(T,K)=SUM(max{0,degT(v)-k})

where degT(v) is the degree of v in T. It is easy to see that for a tree T, T is a k-tree if and only if te(T,k) = 0, and T is a (t+2) -ended tree if and only if te(T,2) t . So using the concept of total excess, we can deal with both spanning k-trees and spanning k-ended trees at the same time.
  In this talk, I will give a survey on spanning trees with bounded total excesses.