We study the broadcast problem on dynamic networks with $n$ processes. The processes communicate in synchronous rounds along an arbitrary rooted tree. The sequence of trees is given by an adversary whose goal is to maximize the number of rounds until at least one process reaches all other processes. Previous research has shown a $\lceil{\frac{3n-1}{2}}\rceil-2$ lower bound and an $O(n\log\log n)$ upper bound. We show the first linear upper bound for this problem, namely $\lceil{(1 + \sqrt 2) n-1}\rceil \approx 2.4n$. Our result follows from a detailed analysis of the evolution of the adjacency matrix of the network over time.