We present bijections connecting tree records, the girth of a connected endofunction, and the genesis sequence (the first sequence in OEIS). Using these, we derive generating functions for tree and forest record numbers in terms of Cayley's tree function and give a new proof of Cayley's forest formula.

翻译：我们建立了连接树记录、连通内函数的围长以及创生序列（OEIS中的首个序列）之间的双射。利用这些双射，我们推导出了用凯莱树函数表示的树与森林记录数的生成函数，并给出了凯莱森林公式的一个新证明。