On the SIG dimension of trees under $L_{\mathrm{\infty }}$ metric

We study the $SIG$ dimension of trees under $L_{\mathrm{\infty }}$ metric and answer an open problem posed by Michael and Quint (Discrete Applied Mathematics: 127, pages 447-460, 2003). Let $T$ be a tree with atleast two vertices. For each $v\in V(T)$, let leaf-degree$(v)$ denote the number of neighbours of $v$ that are leaves. We define the maximum leaf-degree as $\alpha(T) = \max_{x \in V(T)}$leaf-degree$(x)$\mathrm{.}Let$S = {v\in V(T) |$leaf-degree$(v) = \alpha}$. If $\mathrm{\mid }S\mathrm{\mid }=1$, we define $\beta (T)=\alpha (T)-1$. Otherwise define $\beta(T) = \alpha(T)$\mathrm{.}Weshowthatforatree$T$,$SIG_\infty(T) = \lceil \log_2(\beta

• 2)\rceil$where$\beta = \beta (T)$,provided$\beta$isnotoftheform$2^k

• 1$,forsomepositiveinteger$k \geq 1$\mathrm{.}If$\beta = 2^k - 1$,then$SIG_\infty (T) \in {k, k+1}$. We show that both values are possible.