Say I have a directed network that is organized in a tree, with all connections going downstream (genealogically). By that I mean that there is one root node connected to $c_{00}$ child nodes, and each $jth$ child node at generation $i$ (of which there are $\sum_{j} c_{(i-1),j}$) is connected to $c_{ij}$ child nodes. If I line up these $c_{ij}$ as a matrix where $i$ is the column index and $j$ is the row index, I have something like this:
$$ \begin{bmatrix} 2 & 2 & 1 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} $$
This would correspond to a network where the root node is connected to two child nodes, one of which is connected to two grandchild nodes, while the other has no children. The grandchildren are each connected to one child node, both of which are childless. (These values are the Hamming distance of each $ij$th column of the adjacency matrix from zero (the number of nonzero connections at each node).)
I think that the adjacency matrix of a network like this is nilpotent, as long as there are a finite number of generations. Is this conjecture true, or are there limiting conditions on the child counts, or am I totally wrong?