Let $F$ be a field and $L\slash F$ be a finite Galois extension. Let $T_{L\slash F}\colon =R^1_{L\slash F}\Bbb{G}_m$ denote the corresponding norm 1 torus, which is an algebraic group over $F$. Then, in the case when $F$ is a number field, it is a classical result that the local-global principle, or Hasse principle, holds for $T_{L\slash F}$. In other words, the map $$ H^1(F,T_{L\slash F})\to \prod_{v}H^1(F_v,T_{L\slash F}) $$ is injective. \par The paper under review considers a similar question by taking $F$ to be a {\it semi-global field}, i.e., the function field of a smooth, projective, geometrically integral curve over a complete discretely valued field $K$. Then the main result of this paper can be stated as follows: \par Theorem. Let $F$, $L$ and $K$ be as above. Let $\Cal{X}$ be a regular proper model of $F$ such that the reduced special fibre $X$ is a union of regular curves with normal crossings. Suppose that the graph of $\Cal{X}$ is a tree and the residue field $\kappa$ of $K$ is either algebraically closed of characteristic coprime to $[L\colon F]$ or a finite field of characteristic coprime to $[L\colon F]$ that contains a primitive $[L\colon F]$-th root of unity. Then the local-global principle holds for $T_{L\slash F}$. \par In fact, the author proves the above theorem by showing a more general and technical result involving the $R$-trivial elements (cf. §2.3). At the end of the article, several counterexamples to the local-global principle for certain $T_{L\slash F}$ and multinorm tori with $F$ semi-global are given.