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We investigate space-time trade-offs for population protocols in sparse interaction graphs. In complete interaction graphs, optimal space-time trade-offs are known for the leader election and exact majority problems. However, it has remained open whether other graph families exhibit similar space-time complexity trade-offs, as existing lower bound techniques do not extend beyond highly dense graphs. In this work, we show that - unlike in complete graphs - population protocols on bounded-degree trees do not exhibit significant asymptotic space-time trade-offs for leader election and exact majority. For these problems, we give constant-space protocols that have near-optimal worst-case expected stabilisation time. These new protocols achieve a linear speed-up compared to the state-of-the-art. Our results are based on two novel protocols, which we believe are of independent interest. First, we give a new fast self-stabilising 2-hop colouring protocol for general interaction graphs, whose stabilisation time we bound using a stochastic drift argument. Second, we give a self-stabilising tree orientation algorithm that builds a rooted tree in optimal time on any tree. As a consequence, we can use simple constant-state protocols designed for directed trees to solve leader election and exact majority fast. For example, we show that "directed" annihilation dynamics solve exact majority in O(n² log n) steps on directed trees.