Reconstructing a Monthly problem: tree growth on the 2D integer lattice

I think I got it. It seems to be problem 10360, originally published in 1994 (volume 101, issue 1, page 76), proposed by Richard Stanley. I'm looking at the JSTOR page of the solution (by Robin Chapman), which appeared in October 1998, volume 105, no 8, pages 769-771).

Here is the statement of the problem:

Let $L$ be the integer lattice in $\mathbb{R}^d$, i.e., the $L$ is the set of points $(x_1,x_2,\ldots,x_d)$ with all $x_j\in\mathbb{Z}$. Consider $L$ as a graph by declaring two lattice points to be adjacent if the distance between them is $1$. Define a sequence $S_0$, $S_1,\ldots$ of subsets of $L$ inductively as follows: \begin{align*} S_0 &= \Bigl\{ (0,0,\ldots,0)\Bigr\}\\\ S_{n} &= \Bigl\{ P\in L-\mathop{\cup}\limits_{0\leq k\lt n}S_k\ \Bigm|\ \text{$P$ is adjacent to exactly one element of $\mathop{\cup}\limits_{0\leq k\lt n}S_k$}\Bigr\}. \end{align*} Let $S$ be the subgraph of $L$ whose vertices are $\cup S_n$. Thus, $P\in S$ is adjacent to $P'\in S$ if the distance between $P$ and $P'$ is $1$.

1. Find a simple condition for a point of $L$ to belong to $S$.
2. For $P\in S$, find a simple rule to determine $i$ such that $P\in S_i$.
3. How many elements are in $S_i$?
4. How many $P\in S_i$ are adjacent to no points in $S_{i+1}$?
5. Show that $S$ is a tree.
6. Investigate the (vertex) density of $S$ in $L$, and compare it to the largest density of a subset of $L$ for which the induced subgraph is a tree.