I am looking for help:

Beginning with a single node ($\circ$), at each discrete time step I can add a node/link pair to any node currently in the tree. Nodes are unlabelled and the tree is undirected. This will produce a "tree trajectory", like this: $$ \circ \quad \rightarrow\quad \circ - \circ \quad \rightarrow\quad \begin{array}{c} \ \ \ \ \circ - \circ \\ | \\ \circ \end{array} \quad \rightarrow\quad \begin{array}{c} \ \ \ \ \circ - \circ \\ | \\\ \ \ \ \circ - \circ \end{array} $$ Another tree trajectory might look like this: $$ \circ \quad \rightarrow\quad \circ - \circ \quad \rightarrow\quad \begin{array}{c} \ \ \ \ \circ - \circ \\ | \\ \circ \end{array} \quad \rightarrow\quad \begin{array}{c} \circ - \circ \\ | \ \ \backslash \\ \circ \ \ \ \circ \end{array} $$

My questions are as follows:

1) How many possible distinct (non-isomporhic) graphs are there with N nodes and N-1 edges?

It's easy to find this by brute force up to 5: N=1: 1; N=2:1; N=3:1; N=4:2; N=5:3

2) For N nodes, how many different tree trajectories are there connecting the graph with N=1 node to the any graph of N nodes? (i.e., how many distinct tree trajectories are there; a tree trajectory is distinct from all others if for any other tree trajectory, there is at least a single step where the two are not isomorphic.) It's clear not just multiplying the number of non-ismorphic graphs at each step, since past graph topology will constrain future graph topologies.

While a solution would obviously be great, ANY help at all would be appreciated.