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 $\alpha\in [1,2]$, there exist caterpillars with $k$ vertices whose weak saturation numbers are of order $k^\alpha$. 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.