  We apply the methods of algebraic reliability to the study of percolation on
trees. To a complete $k$-ary tree $T_{k,n}$ of depth $n$ we assign a monomial
ideal $I_{k,n}$ on $\sum_{i=1}^n k^i$ variables and $k^n$ minimal monomial
generators. We give explicit recursive formulae for the Betti numbers of
$I_{k,n}$ and their Hilbert series, which allow us to study explicitly
percolation on $T_{k,n}$. We study bounds on this percolation and study its
asymptotical behavior with the mentioned commutative algebra techniques.
