We consider a new pursuit-evasion problem on trees where a subset of vertices, called sources, are initially occupied by searchers. We also consider the scenario where some of the searchers must end their search at certain vertices called targets. We incrementally consider such problems, first considering only sources, then only targets, and finally we consider the case where there are both sources and targets. For each case we provide a polynomial-time algorithm for computing the search number, i.e. the minimum number of searchers required to clear the tree, and an optimal search strategy. We also demonstrate that each search model is monotonic, i.e. for, each case their exists an optimal search strategy such that the set of cleared edges grows monotonically as the search progresses.
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