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 $S(50)$.
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