The Definition is:
BASIS STEP: A single vertex r is a rooted tree.
RECURSIVE STEP: Suppose that T1 ,T2 ,…,Tn are disjoint rooted trees with roots
r1, r2 , …, rn, respectively. Then the graph formed by starting with a root r, which is not
in any of the rooted trees T1,T2,…,Tn, and adding an edge from r to each of the vertices r1, r2 , …, rn , is also a rooted tree.
And it put a picture to illustrate some of the rooted trees formed starting with the basis step and applying the recursive step one time and two times.
I cannot get any idea to connect the definition and the illustrated picture.
Because, according to the definition, we have only one element of the set of rooted trees when we are in basis step, that is, the rooted tree with only one vertex. And then we applied recursive step one time, we should only produce one new rooted tree to the set, which were formed like the first one in the step 1 in the illustrated picture.
Why there were four, even more, new rooted tree produced with applying recursive step only one time? Where does I misunderstand?
Note that there are choices to make in the recursive step: every integer $n$ and every choice of rooted trees $T_1,\dots,T_n$ gives a new rooted tree at that step. Even at step $1$, where the only choice of each $T_j$ is a single vertex, there are infinitely many positive integers $n$ that result in infinitely many new rooted trees at step 1 (the illustration shows $n=1,2,3,4$).