On L(d, 1)-labelling of trees

Given a graph ▫$G$▫ and a positive integer ▫$d$▫, an ▫$L(d,1)$▫-labelling of ▫$G$▫ is a function ▫$f$▫ that assigns to each vertex of ▫$G$▫ a non-negative integer such that if two vertices ▫$u$▫ and ▫$v$▫ are adjacent, then ▫$|f(u)-f(v) |ge d$▫ and if ▫$u$▫ and ▫$v$▫ are at distance two, then ▫$|f(u)-f(v)| ge 1$▫. The ▫$L(d,1)$▫-number of ▫$G$▫, ▫$lambda_d(G)$▫, is the minimum ▫$m$▫ such that there is an ▫$L(d,1)$▫-labelling of ▫$G$▫ with ▫$f(V) subseteq {0,1,dots , m}$▫. A tree ▫$T$▫ is of type 1 if ▫$lambda_d(T) = Delta+d-1$▫ and is of type 2 if ▫$lambda_d(T) ge Delta+d$▫. This paper provides sufficient conditions for ▫$lambda_d(T)=Delta+d-1$▫ generalizing the results of Wang [W. Wang, The ▫$L(2,1)$▫-labeling of trees, Discrete Appl. Math. 154 (2006) 598-603] and Zhai, Lu, and Shu [M. Zhai, C. Lu and J. Shu, A note on ▫$L(2,1)$▫-labeling of Trees, Acta. Math. Appl. Sin. 28 (2012) 395-400] for ▫$L(2,1)$▫-labelling