A peerless tree is a tree with no edge between two vertices of the same degree. Let $P(n)$ be the number of peerless trees on $n$ unlabelled vertices.

There are six of these trees on seven unlabelled vertices, $P(7)=6$, shown below.

Define ${\displaystyle S(N)=\sum _{n=3}^{N}P(n)}$. You are given $S(10)=74$.

Find $S(50)$.