Matrix generators for the unit groups of $L_K(1,d)$

Let $K$ be a field and put $L_d=L_K(R_d)\cong L_K(1,d)$. Ordered leaf sets in the rooted $d$-ary tree determine copies of general linear groups over $K$ inside $L_d^\times$. We prove that these copies generate $L_d^\times$ for every $d\geq2$. In the binary case, $L_2^\times=\langle 1+eaf^*,1+fbe^*:a,b\in L_2\rangle$. We characterize finite generation of $L_d^\times$, determine the subgroup represented by monomial matrices, and embed $\GL_\infty(K)$ in $L_2^\times$. Over a finite field, finite presentability of $L_d^\times$ is equivalent to finite generation of the unstable $K_2$-group $K_2(n,L_d)$ for every $n=1+r(d-1)\geq5$, where $r\geq0$; we also compute $K_2(L_d)$.