A note on weak saturation number of trees

In this paper, we estimate the weak saturation numbers of trees. As a case study, we examine caterpillars and obtain several tight estimates. In particular, this implies that for any α ∈ [ 1 , 2 ] , there exist caterpillars with k vertices whose weak saturation numbers are of order k α . We call a tree good if its weak saturation number is exactly its edge number minus one. We provide a sufficient condition for a tree to be a good tree. With the additional property that all leaves are at even distances from each other, this condition fully characterizes good trees.