Embedding trees of diameter four is NP-hard
It is reducible from Independent Set in cubic graphs.

Given a cubic graph $G(V,E)$, construct a bipartite graph $\mathcal{B}(V\dot{\cup}E,\mathcal{E})$, where $$\mathcal{B}(\mathcal{E})=\{(u,e)|u\in V\ \mathrm{is\ incident\ to}\ e\in E\}$$

Create a new vertex $r$ (the root), connect it to all $V$-vertices in $\mathcal{B}$.

Now, if the IS instance asks for an independent set of size $k$, we create $T$ of diameter $4$ as follows:

Make a new vertex $r_T$ as root of $T$. This vertex would have $k$ children. Each of these $k$ children would have $3$ leaves attached to it.

Our reduction then outputs the instance $\mathcal{B},T$ of $\mathrm{TREE\ EMBEDDING}$ problem.

Note that $k&gt;10^9$, otherwise the vertex cover can be decided in polynomial time. So, if there exists an embedding, $r_T$ must be mapped to $r$.

Since all the leaves are pairwise distinct. The image set of $k$ children of $r_T$ is the independent set.