Number of compatible trees with an ancestry matrix Suppose you are given an ancestry matrix $M$ which means that $M[ij] = 1$ iff node $i$ is an ancestor of node $j$. If $M$ represents no cycles (treated as an adjacency matrix) the corresponding graph is a tree (or a forest). My question is what is the number of trees where their ancestry matrix is $M$. 

Is this question any simpler than counting number of directed graphs which are compatible to a general ancestry matrix?
As hinted in the comments by @j_random_hacker, ancestry matrix, if valid, characterizes a unique rooted tree. A simple algorithm would be to create the tree from leaves to the root or vice versa.