Let G be a graph with diameter d. A radio labelling of G is a function f that assigns to each vertex with a non-negative integer such that the following holds for all vertices $u,v$: $|f(u)-f(v)|⩾d+1-d(u,v)$, where $d(u,v)$ is the distance between u and v. The span of f is the absolute difference of the largest and smallest values in $f(V)$. The radio number of G is the minimum span of a radio labelling admitted by G. For trees, a general lower bound of the radio number was given by Liu [19], which has been used to prove special families of trees whose radio number is equal to this bound [1], [13], [18], [19]. In this article, we present improved lower bounds for some trees whose radio number exceeds Liu's lower bound. Some of these new bounds are sharp for special families of trees, including complete binary trees [18] and odd paths [20]. Moreover, using these new bounds, we extend the known results of Halasza and Tuza [13] on complete level-wise regular trees.

令G为直径为d的图。G的无线电标记是一个函数f，该函数使用非负整数分配给每个顶点，使得对于所有顶点都遵循以下规则$ü，v$： $|F（ü）-F（v）|⩾d+\mathrm{1个}-d（ü，v）$，在哪里 $d（ü，v）$是u和v之间的距离。f的跨度是的最大值和最小值的绝对差$F（V）$。的无线电数ģ是由接受的放射性标记的最小跨度ģ。对于树木，Liu [19]给出了无线电编号的一般下界，这已被用来证明其无线电编号等于该界限的特殊树种[1]，[13]，[18]，[ 19]。在本文中，我们提出了一些树的下限改进，这些树的无线电数超过了Liu的下限。这些新界限中的某些界限对于特殊的树木科来说非常明显，包括完整的二叉树[18]和奇数路径[20]。而且，使用这些新的边界，我们在完整的水平规则树上扩展了Halasza和Tuza [13]的已知结果。