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We prove that for any finite tree $T$ with $n$ vertices and maximal degree$3$, there is a topological embedding of $T$ into the integer grid $Z^2$ whichmaps vertices to vertices and whose image meets at most $\frac{7}{3}n$vertices. This resolves question $7.7$ of arXiv:2112.05305, and furthermoregives the first example of a pair of graphs $X,Y$ such that there is no regularmap $X\to Y$ but the coarse wiring profile of $X$ into $Y$ grows linearly.

A Topological Embedding of the Binary Tree into the Square Lattice