For a vertex set $v = \left\{v_1...v_n \right\}$ and a common root $r$, is there an efficient (maybe $O(1)$ per tree) algorithm that generates all non-isomorphic trees on all nodes $v$ and with root $r$.
Two trees are isomorphic if all the parent-child relationships are the same, i.e. all equivalent nodes in the two trees have the same parent and the same children.
Example: $v = \left\{v_1, v_2, v_3\right\}$
All trees should have the same root an the same node set. The following image shows some valid trees:
Including a tree should lead to all other isomorphic trees being ignored
Another SO post shows a implementation that finds unrooted topologies:
In one answer, there appears this algorithm for creating unrooted non-isomorphic topologies on $n$ nodes in constant time per tree.
Robert Alan Wrights, Bruce Richmond, Andrew Odlyzko and Brendan D. Mckay (1986). “Constant time generation of free trees”. SIAM J. Comput. 15:2, pp. 540–548.
I am not sure if it is possible to extend this algorithm efficiently to generate the desired rooted trees.
I can do it in $O(n^2)$ per tree.
The number of possible trees is $(n+1)^{n-1}$, since choosing a tree in your problem is equivalent to choosing an undirected tree on the vertex set $v\cup \{r\}=\{v_1,\dots,v_n,r\}$, and then directing all edges towards the root. The number of undirected trees on $n+1$ vertices is $(n+1)^{n-1}$ (this is Cayley's theorem), and these trees can be generated efficiently using Prüfer codes.
Specifically, iterate through all lists of length $n-1$ with entries in $\{0,1,2,\dots,n\}$, where $0$ represents $r$ and $1$ through $n$ represent $v_1$ through $v_n$. For each list, generate the corresponding tree with vertices $\{r,v_1,\dots,v_n\}$ using the algorithm to convert a Prüfer sequence into an undirected tree. Finally, turn this into a directed tree using breadth first search starting from vertex $r$.