Abstract

     A tournament of order n is an orientation of a completed 
graph of order n. A rooted tree is a directed tree such that 
every vertex except the root has in-degree 1, while the root 
has in-degree 0. A rooted k-tree is a rooted tree such that 
every vertex except the root has out-degree at most k. It is 
well-known that every tournament has a rooted spanning tree 
of depth at most 2. We proved that every tournament has a 
rooted spanning 2-tree of depth at most 2. This new result 
has many applications in the structure of tournaments.  Most 
of the work was jointly done with Da-Wei Wang and C. K. Wong.