  We concentrate on some recent results of Egawa and Ozeki [J. Graph Theory,
2015 and Combinatorica, 2014], and He et al. [J. Graph Theory, 2002]. We give
shorter proofs and polynomial time algorithms as well.
  We present two new proofs for the sufficient condition for having a spanning
tree with prescribed lower bounds on the degrees, achieved recently by Egawa
and Ozeki. The first one is a natural proof using induction, and the second one
is a simple reduction to the theorem of Lov\'asz. Using an algorithm of Frank
we show that the condition of the theorem can be checked in time
$O(m\sqrt{n})$, and moreover, in the same running time -- if the condition is
satisfied -- we can also generate the spanning tree required. This gives the
first polynomial time algorithm for this problem.
  Next we show a nice application of this theorem for the simplest case of the
Weak Nine Dragon Tree Conjecture, and for the game coloring number of planar
graphs, first discovered by He et al.
  Finally, we give a shorter proof and a polynomial time algorithm for a good
characterization of having a spanning tree with prescribed degree lower bounds,
for the special case when $G[S]$ is a cograph, where $S$ is the set of the
vertices having degree lower bound prescription at least two. This theorem was
proved by Egawa and Ozeki in 2014 while they did not give a polynomial time
algorithm.
