If we have a large (directed) graph $G$ and a smaller rooted tree $H$, what is the best known complexity for finding subgraphs of $G$ isomorphic to $H$? I am aware of results for subtree isomorphism where both $G$ and $H$ are trees and also where $G$ is planar or has bounded treewidth (and others) but not for this graph and tree case.
The question whether any fixed graph $H$ is an (induced) subgraph of $G$ is a first-order definable property, i.e., for every $H$ there is a formula $\varphi_H$ ($\psi_H$) such that $H$ is an (induced) subgraph of $G$ if and only if $G \models \varphi_H$ ($G\models\psi_H$).
It was formerly known that the model-checking problem is fixed-parameter tractable on classes of graphs that (locally) exclude a minor and on classes of (locally) bounded expansion. Recently, Grohe, Kreutzer and S. anounced an even more general meta-theorem, stating that every first-order property can be decided in almost linear time on nowhere dense classes of graphs.
For your question this implies the following. Let $H$ be a fixed rooted tree. Then it can be decided in linear time whether $H$ is an (induced) subgraph of an input (directed or undirected) graph $G$ if $G$ is planar, or more generally is from a class that excludes a minor or from a class of bounded expansion. The problem can be decided in almost linear time if $G$ is from a class that locally excludes a minor or from a class of locally bounded expansion or most generally, $G$ is from a nowhere dense class of graphs.
It can be solved in randomized expected time $O(2^km)$ where $k$ is the size of the small directed tree to be found and $m$ is the number of edges of the large directed graph in which to find it. See Theorem 6.1 of Alon, N., Yuster, R., and Zwick, U. (1995). Color-coding. J. ACM 42(4): 844–856. Alon et al. also state that their algorithm can be derandomized but don't give the details for that part; I think the deterministic time may be a little larger, something more like $O(k!\,m)$.
You are probably looking for Marx,Pilipczuk work on parameterized complexity of Subgraph Isomorphism. Technically, it covers only undirected graphs, but I think you can adapt the hardness results for trees $H$ easily to rooted trees. The positive results relevant for your problem are already covered by the previous answers.