Exact cooling number for spiders and trees

Determine the exact value of the cooling number CL(T)—the maximum number of rounds in the cooling process where each round selects one new cooling source and cooling spreads to neighbors—for spider graphs (trees with a unique vertex of degree at least three) and, more generally, for arbitrary trees.

Background

Cooling is a dual, slow-spread contagion process: in each round one new source is cooled, then cooling spreads to all neighbors of already cooled vertices; the cooling number CL(G) is the maximum number of rounds until all vertices are cooled.

The paper provides bounds and exact values for certain spiders (for example, when the number of legs relative to leg length satisfies specific inequalities), but a general characterization across all spiders and trees is not yet known.