A formula is provided to
explicitly describe global dimensions of all kinds of tree type
finite dimensional $k-$algebras for $k$ an algebraic closed field.
In particular, it is pointed out that if the underlying tree type
quiver has $n$ vertices, then the maximum of possible global
dimensions is $n-1$.
global dimension, tree type finite dimensional $k-$algebra, quiver
16D40 - Free, projective, and flat modules and ideals [See also 19A13]
16E10 - Homological dimension
16G20 - Representations of quivers and partially ordered sets