1

For $\prod_{i=1}^{+\infty}{(x^i-1)}$, let $c_i$ be the coefficient of $x^i$ after expansion. Then

\[c_i=\begin{cases}\pm 1 &\mbox{if } f(i) \in \mathbb{Z}\\ 0 &\mbox{if } f(i) \notin \mathbb{Z}\end{cases}\]

where

\[f(x)=\frac{\sqrt{24x+1}+1}{6}\]

2

For the intersection of the edge sets of two tree paths, $(a,b) \cap (c,d)$:

let $T_{xy}=\mathrm{LCA}(x,y)$, and let $x$ and $y$ be the two deepest points among $T_{ac},T_{ad},T_{bc},T_{bd}$. Then

1: $x=y$

The intersection is empty.

2: $x \neq y$

The intersection is the path $(x,y)$.