Tree covers of size $2$ for the Euclidean plane

For a given metric space $(P,\phi )$, a tree cover of stretch $t$ is a collection of trees on $P$ such that edges $(x,y)$ of trees receive length $\phi (x,y)$, and such that for any pair of points $u,v\in P$ there is a tree $T$ in the collection such that the induced graph distance in $T$ between $u$ and $v$ is at most $t\phi (u,v)\mathrm{.}$ In this paper, we show that, for any set of points $P$ on the Euclidean plane, there is a tree cover consisting of two trees and with stretch $O(1)\mathrm{.}$ Although the problem in higher dimensions remains elusive, we manage to prove that for a slightly stronger variant of a tree cover problem we must have at least $(d+1)\mathrm{/}2$ trees in any constant stretch tree cover in ${\mathbb{R}}^d$.